Involutions of Kac-Moody groups [Elektronische Ressource] / von Max Horn
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Involutions of Kac-Moody groups [Elektronische Ressource] / von Max Horn

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160 pages
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Involutions of Kac-Moody GroupsVom Fachbereich Mathematikder Technischen Universität Darmstadtzur Erlangung des Grades einesDoktors der Naturwissenschaften(Dr.rer.nat.) genehmigteDissertationvonDipl.-Math. Max Hornaus Darmstadt1. Referent: PD dr. Ralf Gramlich2.t: Prof. Dr. Bernhard Mühlherr3. Referent: Prof. Dr. Karl-Hermann NeebTag der Einreichung: 18. Dezember 2008Tag der mündlichen Prüfung: 17. April 2009Darmstadt 2009D 17iiCONTENTSIntroduction vii1. Preliminaries 11.1. Coxeter systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Roots and root systems . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. Involutions and twisted involutions of Coxeter groups . . . . . . . . . 31.4. Chamber systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5. Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6. Twin Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7. BN-pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8. Twin BN-pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9. Root group systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.10.Moufang sets and pointed Moufang sets . . . . . . . . . . . . . . . . . 132. Flips 172.1. Building flips and BN-flips . . . . . . . . . . . . . . . . . . . . . . . . 182.2. Correspondence between building and BN-flips . . . . . . . . . . . . 262.3.

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Informations

Publié par
Publié le 01 janvier 2009
Nombre de lectures 37
Langue Deutsch
Poids de l'ouvrage 1 Mo

Exrait

InvolutionsofKac-MoodyGroups

VomFachbereichMathematik
derTechnischenUniversitätDarmstadt
einesGradesdesErlangungzurhaftenNaturwissenscderDoktorsgenehmigtenat.)rer.(Dr.

Dissertation

onvDipl.-Math.HornMaxDarmstadtaus

t:Referen1.t:Referen2.t:Referen3.TagderEinreichung:
TagdermündlichenPrüfung:

PDdr.RalfGramlich
Prof.Dr.BernhardMühlherr
Prof.Dr.Karl-HermannNeeb
18.Dezember2008
2009April17.

Darmstadt200917D

ii

CONTENTS

ductionIntroriesPrelimina1.1.1.Coxetersystems..............................
1.2.Rootsandrootsystems..........................
1.3.InvolutionsandtwistedinvolutionsofCoxetergroups.........
1.4.Chambersystems.............................
1.5.Buildings.................................
1.6.TwinBuildings..............................
1.7.BN-pairs.................................
1.8.TwinBN-pairs..............................
1.9.Rootgroupsystems............................
1.10.MoufangsetsandpointedMoufangsets.................
Flips2.2.1.BuildingflipsandBN-flips........................
2.2.CorrespondencebetweenbuildingandBN-flips............
2.3.Steepdescent...............................
2.4.Strongflips................................
2.5.Stabletwinapartments..........................
2.6.2-divisiblerootgroups..........................
2.7.Doublecosetdecomposition.......................
3.Flipsinrank1and2
3.1.FlipsofSL(F)andPSL(F).......................
223.1.1.ClassifyingflipsofSL(F)....................
23.1.2.Centralizersofflips........................
3.2.FlipsofMoufangsets...........................
3.3.Classicalquadrangles...........................
3.3.1.SomeauxiliaryresultsonMoufangsets.............
3.3.2.Commonsetting.........................
3.3.3.Directdescent...........................
θ3.3.4.Risalmostalwaysconnected..................

vii11234679101131171826280331353841414244454747505255

iii

tstenCon

4.Structureofflip-flopsystems61
4.1.Flip-flopsystems.............................61
4.2.Outlineoftheproof............................64
4.3.MinimalPhanresidues..........................65
4.4.Homogeneityandinheritedconnectedness...............67
4.5.Residualconnectedness..........................69
4.6.Rank2residues..............................71
4.6.1.Risaθ-orthogonalrank2residue................73
4.6.2.Risaθ-acuterank2residue...................73
4.6.3.Risaθ-parallelrank2residue.................76
4.7.Statementofthemaintheorems.....................82
5.Transitiveactionsonflip-flopsystems85
5.1.Transitivity................................85
5.2.Alocalcriterionfortransitivity.....................90
5.3.Transitivityinrank1...........................91
5.3.1.Transitivityinrank1:SLandPSL..............92
225.3.2.Transitivityinrank1:Moufangflips..............95
5.4.Iwasawadecompositions.........................97
5.4.1.FieldsadmittingIwasawadecompositions............101
5.5.Moreonflipsoflocallysplitgroups...................102
6.ApplicationstoalgebraicandKac-Moodygroups105
6.1.Algebraicgroups.............................105
6.1.1.Quasi-flipsofalgebraicgroups..................106
6.1.2.Applicationstoalgebraicgroups.................107
6.1.3.Linearflips............................108
6.1.4.Semi-linearflips..........................108
6.2.GroupsofKac-Moodytype.......................109
6.2.1.LocallyfiniteKac-Moodygroups................111
115resultsComputerA.θA.1.ConnectednessofR:θ-acutequadrangles...............115
θA.2.ConnectednessofR:θ-parallelprojectiveplanes...........115
θA.3.ConnectednessofR:θ-parallelquadrangles..............116
A.4.GAPcode.................................118
B.Openproblems125
C.PhantheoryusingMoufangsets129
133Bibliography143Index

iv

ZusammenfassungheDeutscHistorischsindInvolutionenzweifellosvongroßemInteresse,beispielsweiseimRah-
menderKlassifikationderendlicheneinfachenGruppen(inwelcherZentralisatoren
vonInvolutioneneinegroßeRollespielen)oderzurDefinitionvonsymmetrischen
RiemannschenRäumenbzw.vonsymmetrischenk-Varietäten.Zieldervorliegen-
denArbeitistdasStudiuminvolutorischerAutomorphismenreduktiveralgebraischer
GruppenundzerfallenderKac-Moody-Gruppen(indiesemFallsolldieInvolution
diebeidenKonjugiertenklassenvonBoreluntergruppenvertauschen)inCharakteris-
tikungleich2,sowiederenZentralisatoren.
DiegenanntenGruppenhabengemein,dasssiezueinemZwillingsgebäudeassozi-
iertsind.SeiGnuneinesolcheGruppe.EininvolutorischerAutomorphismusθvon
Ginduzierteinenfast-isometrischenAutomorphismusdesassoziiertenGebäudesC.
Diesermöglichtes,diereichhaltigeStrukturtheorievonGebäudenanzuwenden.
EinwichtigesHilfsmittelhierbeiistdassogenannteFlipflop-SystemCθ,bestehend
ausallenKammernderpositivenHälftedesGebäudes,welchedurchdieinduzier-
teAbbildungθmaximalweitabgebildetwerden(imSinnederKodistanzaufθdem
ZwillingsgebäudeC).AlsTeilkammernsystemdesGebäudesC+kannmanCauch
alssimplizialenKomplexauffassen.DerZentralisatorGθvonθinGwirktaufdiesem
omplex.KSeiGeineGruppemitZwillings-BN-Paar(B+,B−,N)undZwillingsgebäude
C=(C+,C−,δ∗)undθeine(fast-)isometrischeInvolutionvonC.Dieursprüngli-
cheMotivationfürdievorliegendeArbeitbeinhaltetdieBeantwortungderfolgenden
Fragen,welchesunsimWesentlichengelungenist:
•WannkannmanθzueinerInvolution(oderwenigstenseinembeliebigenAu-
tomorphismus)derGruppeliften?
•WannistCθalsKammernsystemzusammenhängend?
•WannistCθeinreinerSimplizialkomplex?Äquivalent,wannistCθKammern-
metrie?Inzidenzgeoeinersystem•Wennθ∈Aut(G)ist:WannwirktderZentralisatorGθtransitivaufCθ?All-
gemeiner,waskönnenwirüberdieBahnstrukturaussagen?
•WannistGθendlicherzeugt?
•WennCθundC+übereinstimmenundGθtransitivwirkt,erhaltenwireine
verallgemeinerteIwasawa-ZerlegungG=GθB+.Wannistdiesmöglich?
Abschließendseierwähnt,dasssichunsereResultateaufweitereGruppenmitei-
nemWurzelgruppendatumimSinnevon[Tit92](wiez.B.endlicheGruppenvom
Lie-Typ)erweiternlassen.IndiesemFallmussdieKlassederbetrachteteninvolut-
orischenAutomorphismenleichteingeschränktwerdenmitderForderung,dasseine
einzelnegewählteBoreluntergruppeBwiederaufeineBoreluntergruppeabgebildet
wird(imFallevonKac-Moody-GruppenaufeinemitentgegengesetztemVorzeichen).
WirsprechendannvoneinemQuasiflipundbezeichnendamitsowohldieAbbildung
aufderGruppewieauchdieaufdemGebäude.

tstenCon

vi

ODUCTIONINTR

Inthisthesiswestudyinvolutoryautomorphismsofreductivealgebraicandsplit
Kac-Moodygroupsoverarbitraryfields,ormoregenerally,ofgroupswitharoot
groupsystem,asdefinedbyTits[Tit92](thisincludesalsofinitegroupsofLietype,
example).forTheunifyingaspectofallthesegroupsisthattoeachofthematwinbuildingis
associated.ItturnsoutthatanyinvolutoryautomorphismθofagroupGaslisted
abouniquevewainducesy.Wanecallalmosttheseinisometricvolutoryautomorphismautomorphismsofthe(botassohofciatedthegroupbuildingandCinthea
.flipsquasibuilding)Thiscorrespondenceisthekeyinsightdrivingthepresentwork.Wecannowex-
ploitderivetheproprichertiestheoryoftheofbuildingbuildingsinautomorphismgeneraland–andoftwinaccordinglybuildings,viaintheparticularcorrespon-to
dencewehintedatabove,alsooftheoriginalinvolutoryautomorphismθ.Wewill
sketchsomeoftheresultsinwhatfollows.

historySomeButfirst,some“historical”background:Inhindsight,thestudyofflips(aspecial
caseofourflips,wherethebuildingmorphismistypepreserving)wasinitiatedinthe
revisionofthePhantheoremsduetoKok-WeePhan(see[Pha77a]and[Pha77b]).
Theseplayacentralroleintheclassificationoffinitesimplegroups.1Duringthis
effortofreprovingandextendingPhan’stheorems,dubbedalso“Phanprogram”,a
seriesofpublicationswasstartedtoreproveandextendtheclassificationtheorems
byPhan.Theoriginalproofswererathernon-conceptualandinvolvedheavycalcu-
lationsinunitarygroupsandwithgeneratorsandrelations,whichoftenwereeven
onlyalludedtobeomitted.Intherevisedprogram,ageometricapproachwasused
instead,wherethegroupsinquestionweredescribedascentralizersofinvolutions–
involutionswhichwetodaywouldcallflips.
ForanoverviewofthegeneralPhanprogram,wereferto[BGHS03]andalsomore
recently[Gra].ThecaseAnwasdealtwithin[BS04],thecaseBnin[BGHS07]and
[GHN07],2thecaseCnin[GHS03],[Gra04],[GHN06]and[Hor05],thecaseDnin
[GHNS05].1Phan’sresultsenteredtheclassificationviaAschbacher’spaper[Asc77].
2TheA3=D3casealsoleadto[Hor08],whereaspecificexceptiontothePhantheoremsisstudied

vii

ductiontroIn

Initially,duringtheabove-mentionedprogram,somewhat“ad-hoc”choicesofsuit-
ableinvolutoryautomorphismsweremade.Butitsoonbecameapparentthata
deepersystematicreasonwashiddenbelowthesurface.Thisconnectionturnedout
tobebuildingtheory.Allinvolutionsthathadbeenusedcouldbeunderstoodin
termsofthebuildingsoftheinvolvedgroups.Withthisinsight,thegrouprecog-
nitionandpresentationresultsdescribedaboveallfollowveryroughlyanargument
alongthefollowinglines:Givena“targetgroup”G(forwhichwewanttoprovea
recognition/presentationresult),findagroupHendowedwithasphericalBN-pair
andaninvolutoryautomorphismθofHsuchthatGisisomorphictothecentralizer
ofθinH,andsuchthatθalsoinducθesaninvolutoryautomorphismonthe(spherical)
buildingofH.DefineasubsetCofthebuilding(theflip-flopsystem)consistingof
allchambersmappedmaximallyfarawaybyθ.IfonecanshowthatCθisconnected
andsimplyconnected(abuildingisasimplicialcomplexandCθcanbeinterpreted
asasubcomplex),andifmoreoverGactstransitivelyonCθ,thenbyTits’Lemma
(seee.g.[Pas85,Lemma5],[Tit86,Corollary1])thegroupGisfinitelypresented.
ThisinsightfinallymadeitpossibletocarryoutthePhanprograminitsfull
generalityasdescribedabove.Now,therewasaconceptualargumentwhysimple
connectednessandtransitivitywouldsufficetoderivethedesiredresultsongroups.
Therewouldbemuchmoretosayaboutthishistory,butthatisfarbeyondthe
scopeofthisintroduction,sowestopherenow.

GoalsSummarizedandsimplified,thestartingpointofthetheoryofflipswasthestudy
offinitegroupsofLietypebyanalyzing(centralizersof)involutoryautomorphisms
viatheirinteractionwiththesphericalbuildingsassociatedtothegroups.
Thestartingpointofthisthesiswasthedesiretostudyarbitrary“flips”θofsome
reductivealgebraicgroupGwithBN-pair(B+,B−,N)oftype(W,S)withthevague
hopeoflaterextendingthistoKac-Moodygroups.OriginallyaproperBN−flipwas
understoodtobeaninvolutionwhichinterchangestheBorelgroupsB+andB−and
centralizestheWeylgroupW.Thesewouldtheninduceaproperbuildingflipofthe
associatedtwinbuilding,meaningapermutationofthetwinbuildinginterchanging
thetwotwinhalvesisometrically(preservingdistancesandcodistances).Associated
tothisistheflip-flopsystemCθconsistingofallchamberswhicharemappedtoan
oppositechamberbytheflip.
Questionsthatweaskedincluded:WhencanabuildingflipbeliftedbacktoaBN-
flip(theotherdirectionbeingstraightforward)?Whatcanonesayabouttheflip-flop
systemintermsofconnectednessandtransitivitypropertiesofthecentralizerGθof
θinG?Moreover:WhenisCθthechambersystemofanincidencegeometry?Very
earlyon,therewasalsotheideaofgeneralizingIwasawadecompositionsinthevein
[HW93].ofdetail.in

viii

ThisInpapfacter[HW9deals3]withturnedtheoutstudytoofbeinavmaolutorjorysourcealgebraicofinspirationmorphismsofandthemotivgroupation.of
Findeed,-rationalsuchpoinantsofautomorphismconnectedisareductivprimaryealgebraicexampleforgroupsadefined(quasi-)flip!overaMoreofieldver,Fin–
over[KW92]somealgebraicallyresultsclosedsimilarfieldstointhosecinharacteristic[HW93]0butwereapplyinggiven.toOurKac-Mohopeowdyastogroupsuse
buildingtheorytounifyandextendtheseresultsto(almost)arbitraryalgebraicand
groups.dyoKac-MoIntheend,wemanagedtoachievemostofthegoalssketchedaboveandeven
alotbeyondthat:Forexample,insteadofjustalgebraicgroups,wewereableto
also[HW93]coverweKac-Moextendeoddyourgroups,notionoffiniteflipsgroupstoofquasi-flipsLietype(whereandtheotherassumptiongroups.BasedthatWon
ishadcenhoptralizeedtodcanshowbefordropptypeed)andpreservingmanagedpropertoproflipsveofmostalgebraicofthegroupsthingswforearbitoriginallyrary
quasi-flipsofgroupswithatwinBN-pair.
resp.Forallsimplethis,conne[DM07]ctednesshadaofcrucialcertaininfluence.subsetsInofthatbbuildingseautifulispapreduceder,toaconnectednessstudyof
Therank2resultsresp.applyrank3inresiduesparticularviatoantheelegantflip-flopfiltrationsystemsandfromloabocal-to-globalveassociateargumendtots.a
largeclassofinterestinginvolutions(e.g.semi-linearinvolutionsofsplitalgebraicor
Kac-MoodygroupsinterchangingaBorelgroupwithanoppositeone).Hencepartof
the3.3andpresen4.6twethesisshowdealsthatwiththerelevstudyingantcosetsarennectednessindeedintheconnectedrank2incase.“most”InSectcasesionsif
onlysingleordoublebondsexistintheDynkindiagramofthegroup.
Unfortunately,itturnedoutthatnotallinvolutionsweareinterestedinallowfor
a“nice”filtration.Thus,wehadtorefinethestrategyusedin[DM07]andreplace
thecomplicatedsimpleraranknk12proppropertyerty,usedandtheproretovingaestablishsimilarthelorequiredcal-to-globalfiltrationresultbasyainlomorec.
cit.(seeChapter4).AgaininSections3.3and4.6weshowthatthispropertyis
ofsatisfiedthegroup.in“most”casesifonlysingleordoublebondsexistintheDynkindiagram
Thereareseveralaspectsthathavenotyetbeenfullysettled;forexample,we
showhowtoreducethequestionaboutconnectednessoftheflip-flopsystemtoa
imprank2ortantproblem,caseswbutedid,haveandnotytheetbeenremainingabletoaresubhandlejectallofrankongoing2cases.researcStillh.inseveral

thesisthisofStructure1ChapterInthischapter,weintroducemanyoftheconceptsusedthroughoutthepresent
thesis.Itbynomeansattemptstobecomprehensive;ratheritismeanttosettlesome
notationalquestions,introducethefundamentals,andfinallyprovidetheinterested

ix

ductiontroIn

thereaderentirewithwohinrktsisonwher[AB08].etoloThisokforrecenfurtbtherookdetails.presentsOuramaindetailedreferencetreatmenthrtofoughoutthe
theoryrecommendofitbuildings,toevterybwinody,buildings,inparticularandgroupstoreadersactingofonthethem.presenAstwsucork.h,weheartily

2Chapterpair.Here,wTheecloseformallyincorresptroduceondencebquasi-flipsetweenoftthewintwobuildingsconceptsandisgrmadeoupswithprecise.atwinVBariousN-
inOneoftermediatethemostresultsimpareortantcollecteonesdandcertainlyprovenisthethere,followhicwinghare(andheaitsvilygroupusedlatertheoreticon.
terpart):counTheorem1(cf.Theorem2.5.8).Letθbeaquasi-flipofatwinbuilding“notdefined
incharacteristic2”.Thenanychamberciscontainedinaθ-stabletwinapartment.
ThistheoremwasinspiredbycorrespondingworkdonebyAloysiusG.Helminck
ininc[HW93],haracteristicwhereadifferensimilartfromresult2.isHoprowveveder,forthereductivmethoedswalgebraiceemploygroupsareoverbuildingfields
type.theoretic,Inandaddition,thusite.g.canbalsoecapplyonsideredtoasKac-MoaspoecialdycasegroupsofaandmorefinitegeneralgroupsoftheoremLie
provedbyBernhardMühlherrinhisPhDthesis[Müh94];comparedtothattheorem,
however,thepresenttheoremimposesweakerconditionsonGandθ.
Wethenproceedbystudyinginmoredetailwhentherequirementsfortheprecise
versionoftheprecedingtheoremaresatisfied.Thechapterconcludeswithaparam-
eterizationofaninterestingdoublecosetdecomposition.Weonlygivetheversion
foralgebraicandKac-MoodygroupsfromChapter6:
Theorem2(cf.Corollaries6.1.4and6.2.2ofProposition2.7.2).SupposeGisa
cfieldonneFctedwithcisotrharopicF=re2,ductiveandPaalgebraicminimalgrpoup,araborolicaFsplit-subgrKoup.ac-MoLoetdyθbedefineandabstroveracta
twocinvolutoryonjugacyautomorphismclassesofofBorGelgr(intheoups).caseLetof{AKi|ac-Moi∈oIdy}grberoups,eprinteresentativeschangingofthethe
Gθ(F)-conjugacyclassesofθ-stablemaximalF-splittoriinG.Then
Gθ(F)\G(F)/P(F)=∼WGθ(F)(Ai)\WG(F)(Ai).
I∈iThisgeneralizesasimilarstatementforalgebraicinvolutionsgivenin[HW93]
(whic[Mat79],hinturnRossmannwasa[Ros79]generalizationandSpringerofearlier[Spr84]).resultsSeeonalsospecial[KW92]casesforabvyersionMatsukifor
Kac-Moodygroupsoveralgebraicallyclosedfieldsincharacteristic0.

3ChapterInlaterchapters,wefrequentlyperformlocal-to-globalandglobal-to-localargu-
men(Moufangts.Asetsccordinglyand,Moufangunderstandingpolygons)isquasi-flipsofsomeofimpMoufangortance.buildingsofrank1and2

x

InGramlicthishchapter,[DMGH09]wewherefirstwepresenstudytsometransitivitjointwyorkpropwithertiesTofomDequasi-flipsMedtsofandcertainRalf
5rankto1studybuildings,transitivitnamelyypropproertiesjectiveoflinesquasi-flipsoveraofskloewcallyfield.splitThisgroupsisusedandinbuildingsChapter
rank.higherinnessFoftheurthermore,so-calledwestudyflip-flopquasi-flipssystemofisclassstudiedicalandcgeneralizedharacterizedquadrangleforsthese.Connected-buildings.
ThisisthenusedinChapter4.

4ChapterθWeconsistsintroofduceallcthehambflip-flopersofthesystempCositivofeahalfC+quasi-flipwhicθhofareatwinmappedbuildingmaximallyC.Thisfar
away.Tobeprecise,
Cθ:={c∈C+|lθ(c)=d∈Cmin+lθ(d)}.
IfCcomesfromagroupG,andθθcomesfromaquasi-flipofG,thenthecentralizer
GθInofθinChapterG4wnaturallyestudyactstheonC.structureThisisofCforθ.TheexamplekeyusedquestionsinwChaptereinv5.estigateare
whenCθisconnectedasachambersystem,θandwhetheritisresiduallyconnected.
Weobtainedalsobystudyalohomogeneitcal-to-globalypropargumenertiestofandC.theOuracarefulmaintheoremanalysisisofthequasi-flipsfollowing,of
buildings.2rankTheorem3(Theorem4.1.10,jointworkwithGramlichandMühlherr).LetFbe
afieldwithcharF=2andletGbeanisotropicconnectedreductivealgebraicora
G.splitAKssumeac-MoothedygrdiagroupamisdefinedsimplyoverlacFed;andorofassumetype(Wthat,S().W,LSet)θisb2ea-sphericquasi-flipal,Gisof
F-locallysplit,|F|>4,andnoθG2residuesoccur.
Thentheflip-flopsystemCisconnectedandequalstheunionofallminimalPhan
rK-residues,esidueswhichofCθiniscturnonneallctedhaveandridenticesidualallycsphericonnealctetypd.eK.Thechambersystemof
TomotivatewhyweareinterestedinconnectednessofCθ,letusjustmentionthat
itisoneofthekeypointsintheproofofTheorem6.2.5,which,roughlysaid,states
thatGθis“usually”finitelygeneratedifGisalocallyfiniteKac-Moodygroupwith
diagram.-spherical2

5ChapterOncemore,θisaquasi-flipofagroupG,andGθthecentralizerofθinG.Inthis
chapterweturntostudyingtheactionofGθontheflip-flopsystemCθasintroduced
4.Chapterin

xi

ductiontroIn

Themainresultsofthischapterareallbasedontheideaofgeneralizingthe
Iwasawadecompositionofnon-compactconnectedsemi-simplerealLiegroupsto
arbitrarygroupswitharootgroupsystem.Wemakethefollowingdefinition:
Definition(Definition5.4.1).AgroupGwithatwinBN-pair(B+,B−,N)admits
anIwasawadecompositionifthereexistsaninvolutionθ∈Aut(G)whichmaps
B+toB−andsatisfiesG=GθB+,whereGθ:=FixG(θ).
UsingthelocaltransitivityresultsofChapter5,wearriveatthefollowing,which
isoneofthemotivationsforourinterestingeneralizedIwasawadecompositions:
Theorem4(Theorem5.4.2,jointworkwithGramlichandDeMedts).Consider
agroupGendowedwithasystemofrootgroups{Uα}α∈Φwheretherootgroups
generateG(e.g.aKac-Moodygrouporasplitsemi-simplealgebraicgroup),and
withaninvolutionθsuchthatG=GθBisanIwasawadecompositionofG.Fur-
thermore,letΠbeasystemoffundamentalrootsofΦandfor{α,β}⊆Πlet
Xα,β:=Uα,U−α,Uβ,U−β.
ThenθinducesaninvolutiononeachXα,βandGθistheuniversalenveloping
groupoftheamalgam((Xα,β)θ){α,β}⊆ΠoffixedpointsubgroupsofthegroupsXα,β.
WealsocharacterizewhenagroupactuallyadmitsanIwasawadecompositionin
oursense.WegivetheversionforalgebraicandKac-MoodygroupsfromChapter6:
Theorem5(Corollaries6.1.6and6.2.4ofTheorem5.4.7;jointworkwithGramlich
andDeMedts).LetFbeafieldandletGbeasplitconnectedreductivealgebraicor
splitKac-MoodygroupdefinedoverF.ThegroupofF-rationalpointsG(F)admitsan
IwasawadecompositionG(F)=Gθ(F)B(F)ifandonlyifFadmitsanautomorphism
σoforder1or2suchthat
(1)−1isnotanorm,and
(2)(i)eitherasumofnormsisanorm,or
(ii)asumofnormsisεtimesanorm,whereε∈{+1,−1},(andthiscase
canonlyoccurifallrank1subgroupsofGareisomorphictoPSL2(F)),
withrespecttothenormmapNσ:F→FixF(σ):x→xxσ.
6ChapterHere,wespecializesomeofthekeyresultsoftheprecedingchapterstothecase
ofisotropicreductivealgebraicandsplitKac-Moodygroups,withthehopethatit
ismoreaccessibletoreadersfamiliarwitheitheralgebraicorKac-Moodygroups,
butwithlessofabackgroundinbuildingtheory.Assuch,itisintendedtobe
readableonitsown,withoutexplicitlyrequiringtheknowledgeofpreviouschapters
tounderstandtheresultspresentedthere.
Abovewealreadydescribedsomeoftheresultspresentedinthischapterbutone
moreshouldbementioned:

xii

overTheoremafinite6field(TheoremF,q≥56.2.5)and.oSuppdd,osewithG2isa-sphericsplitalKdiagrac-Moamody(andgroupnoofGtypre(Wesidues).,S)
2qLetθbeaquasi-flipofG,i.e.,aninvolutoryautomorphismofGwhichinterchanges
thetwoconjugacyclassesofBorelgroups.ThenthecentralizerGθofθinGis
d.ategenerfinitelysubThejectofrestrictionresearchthatinnoprogressG2byresidueHendrikmayVturnanupcanMaldeghemprobablyandbethedroppauthored.This[HVM].is

endicesAppInAppendixA,wepresentsomeresultsobtainedwiththehelpofacomputer,as
wellastheprogramcodethatwasused.Theseresultscomplementandcomplete
theanalysisofquasi-flipsofMoufangpolygonsasperformedinChapters3and4.
InAppendixBwepresentalistof(inmyeyes)interestingopenproblemsthat
turnedupwhileworkingonthisthesis.Thesemayserveasinspirationandstarting
pointforfutureresearch.
InAppendixC,wesketchhowtogeneralize[BS04]fromfinitefieldstoarbitrary
fieldsusingthemethodsdevelopedinSection3.3.1.

Acknowledgments
FirstandforemostIwouldliketoexpressmydeepgratitudetowardsmyprimary
advisorRalfGramlich,wholeadmethroughthisproject.IamnotsurewhetherI
wouldhavelastedthroughallthiswithouthisguidanceandconstantsupport.Ralf
taughtmefarmorethanjustmathematics,andalwayssetagreatexampleforall
ts.studenhisFurthermore,Iamindebtedtomysecondadvisor,BernhardMühlherr,whose
influencewasespeciallyessentialinChapter4.DuringtwostaysinBruxellesand
manyfruitfuldiscussions,hehelpedmefurthermyunderstandingofthe“building”
aspectoftheproblemstackledinthisthesis.
IalsowouldliketothankHendrikVanMaldeghem,whotaughtmealotabout
Moufangpolygonsandworkedwithmeonthelocalanalysisoftheflip-flopsystems.
AnextendedversionoftheresultsfoundinSection3.3issubjectofaforthcoming
publication.Thanksalsogototothefollowingpeople:TomDeMedts,whoworkedwith
usontheIwasawaresultsandinparticularontheMoufangsetaspectofthat;I
learnedalotaboutMoufangsetsfromthat.MycolleaguesandfriendsAndreas
MarsandStefanWitzelgavemoralsupportandwerealwaysopenforinteresting
mathematicaldiscussions,butalsoformuchlighterconversation.Aloysius“Loek”
Helminckencouragedmetogoonwiththisproject,andprovidedfurtherinsights
ontheresultsin[HW93].OursecretaryGerlindeGehringwasalwaysthereforme
andhelpedmefocusonmyworkbydoinganoutstandingjobtakingcareofallthe
administrativeissuesthatcroppedupduringmyyearsinDarmstadt.Myparents

xiii

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CHAPTERONE

PRELIMINARIES

Inthischapter,wegiveabriefintroductiontosomeofthekeyconceptsusedthrough-
outthepresentthesisaswellasbibliographicalreferencesforfurtherreading.Note
thatwedonotstrivetobecompleteinourliteratureoverview.Insteadourmain
referencethroughoutthischapter(andconsiderablepartsoftherestofthepresent
thesis)is[AB08],andwetrytoalwaysincludeareferencepointingthere.Con-
sequently,itshouldbepossibletoreadlargepartsofthisthesiswithloc.cit.as
exclusivereference.However,westilltrytoincludemanyoriginalandalternate
references.Almosteverythinginthischapterisstandard,withpossiblytheexceptionofparts
ofSection1.3.Thereaderwhoisalreadyfamiliarwiththeconceptsintroduced
belowiswelcometoskippartsorallofthischapter.

systemsxeterCo1.1.ForageneralintroductiontoCoxetergroupsandCoxetersystems,wereferto[Bou68]
(anditsEnglishtranslation[Bou02]),[Bro89](anditssuccessor[AB08]),[Hum90],
andforasomewhatdifferentapproach,[BB05].
LetGbeagroup.Theorderofanelementg∈Gisdenotedbyo(g).
(calledDefinitionCoxeter1.1.1.AgroupCo)andxeterassetSystem⊂Wisasuchpairthat(W,SS)=W,consistings2=1ofWa=sgroupforallW
s∈SandsuchthatthesetSandtherelators((st)o(st))s,t∈Sconstituteapresentation
.WofExample1.1.2.Letnbeanaturalnumber.LetSbethesetofalltranspositions
(pi,ierm+1)utationsfor1of≤ithe≤nset.{1,Then...,Wn+:=1}S,andis(W,isomorphicS)isatoCothexetersymmetricsystem.groupofall
CoRemarkxetersystem1.1.3.(WA,SCo).xetergroupWingeneraldoesnotuniquelydeterminethe

1

liminariesPre1.

iscalledDefinitionthet1.1.4.ypeofLet(W(,WS,)S.)FboreaanCoxetelemenertwsystem.∈WwTheeputmatrixM(S):=(o(st))s,t∈S
lS(W):=min{k∈N|w=s1s2∙∙∙skwheresi∈Sfor1≤i≤k}.
ThenumberlS(w)iscalledthelengthofw.IfSisclearfromcontext,onecommonly
writesl(w)insteadoflS(w).Awordw=s1∙s2∙∙∙sn,withsi∈S,iscalledreduced
ifl(w)=n.
Definition1.1.5.ForasubsetJofSweputWJ:=J.Thisgroupiscommonly
referredtoastheparabolicsubgroupoftypeJ.
Proposition1.1.6(Theorem5.5in[Hum90]).Thepair(WJ,J)isagainaCoxeter
system,obtainedfromtheoriginalonebyrestrictingStoJ.Forallw∈WJone
haslJ(w)=lS(w).
aninDefinitionteger.Iffor1.1.7.allIfWsubsetsitisJoffinite,Sofwesizecallat(W,mostS),nWtheandCoSxetersphericalsystem.(WLetJ,nJ)bise
spherical,wecall(W,S),WandSn-spherical.
Proposition1.1.8(Section5.5in[Hum90]).AsphericalCoxetersystem(W,S)
admitsauniquelongestelement,i.e.,anelementw∈Wsuchthatl(w)>l(w)
forallw∈W\{wS}.Ingeneral,ifJisasphericalSsubsetofS,thenweSdenotethe
longestelementofWJbywJ.
InCoxetersystems,theExchangeconditionholds:
sarilyTheoremareduce1.1.9dexpr(Theoremession.5.8Suppinoses[Hum90])∈S.Lsatisfiesetwl(=wss1)∙∙<∙ls(rw()s.i∈ThenS),thernoteneiscanes-
indexiforwhichws=s1∙∙∙si∙∙∙sr(omittingsi).Iftheexpressionforwisreduced,
unique.isithenDefinition1.1.10.Let(W,S)beaCoxetersystem.Anautomorphismof(W,S)
isagroupautomorphismofWwhichnormalizesS.

1.2.Rootsandrootsystems
Formoreonrootsystems,wereferto[AB08,AppendixB],[Bou68](anditsEnglish
translation[Bou02]),[CR08,PartI.1],[Wei03,Chapter3],tonameafew.
Let(W,S)beaCoxetersystem.Inaccordancewith[AB08,Section5.5.4],we
wing:follothedefineDefinition1.2.1.Foreachs∈S,thesetαs={w∈W|l(sw)>l(w)}isasimple
rootof(W,S).Arootisasetoftheformw.αs,wherew∈Wandαsisasimple
ot.roLetΠ:={αs|s∈S}bethesetofsimplerootsof(W,S),letΦbethesetofall
rootsof(W,S).

2

1.3.InvolutionsandtwistedinvolutionsofCoxetergroups

Definition1.2.2.Arootα∈Φiscalledpositiveifα=w.αsandl(sw)=l(w)+1;
itiscallednegativeifα=w.αsandl(sw)=l(w)−1.
Onecanshowthateveryrootiseitherpositiveornegative,andthatifα=w.αs
isapositiveroot,then−α:=W\α=ws.αsisanegativeroot.
Forε∈{+,−},letΦεdenotethesetofpositive,resp.negativerootsofΦwith
respecttoΠ.Forarootα∈Φ,denotebysαthereflectionofWwhichpermutesα
and−α.Foreachw∈W,defineΦw:={α∈Φ+|w.α∈Φ−}.
Definition1.2.3.Apair{α,β}ofrootsiscalledprenilpotentifα∩βand(−α)∩
(−β)arebothnonempty.
Inthatcasedenoteby[α,β]thesetofallrootsγofΦsuchthatα∩β⊆γand
(−α)∩(−β)⊆−γ,andset]α,β[:=[α,β]\{α,β}.
1.3.InvolutionsandtwistedinvolutionsofCoxeter
groupsInthemainbodyofthepresentwork,wefrequentlyneedpropertiesofinvolutions
(elementsoforder2)ofCoxetergroups.Infact,weneedtodealwithasomewhat
widerclassofelements,so-calledtwistedinvolutions.Thefollowingisbasedon
[Spr84,Section3](seealso[HW93,Section7]).
Definition1.3.1.Let(W,S)beaCoxetersystemandθanautomorphismof(W,S)
oforderatmost2.Aθ-twistedinvolutioninWisanelementw∈Wwith
θ(w)=w−1.WedenotethesetoftheseelementsbyInvθ(W).
ThusInvId(W)isthesetofallinvolutionsofWintheordinarysense.
Lemma1.3.2.Letw∈Invθ(W)beaθ-twistedinvolution,lets∈Sbearbitrary.
Thenl(sw)=l(wθ(s)).Moreoverifl(swθ(s))=l(w)thensw=wθ(s).
Proof.Sinceθisanautomorphismof(W,S),wehavel(w)=l(θ(w))forallw∈W.
Thefirstequalityfollowsreadily:
l(sw)=l((sw)−1)=l(w−1s−1)=l(θ(w)s)=l(wθ(s)).
Thesecondstatementisaconsequenceof[Spr84,Lemma3.2].Fortheconvenience
ofthereader,hereistheproof,adaptedfromtheonegiveninloc.cit.:
Assumesw<wandl(swθ(s))=l(w).Thenwemaywritew=s1∙∙∙shwith
si∈S,s1=s,andl(w)=h.Thenalsow=θ(w)−1=θ(sh)∙∙∙θ(s1).Sincesw<w,
wehavebytheExchangeconditionthatsw=θ(sh)∙∙∙θ(si)∙∙∙θ(s1)forsomeiwith
1≤i≤h.Ifi>1thenl(swθ(s))<l(w)contradictingourhypothesis.Hencei=1
andswθ(s)=w.
Theproofforsw>wissimilar.Assumeagainw=s1∙∙∙shwithsi∈S,and
l(w)=h.Thensw=ss1∙∙∙sh.Byhypothesis,wehavel(swθ(s))=l(w)<l(sw).
ThereforetheExchangeconditionimpliesthatswθ(s)=s1∙∙∙sh=w.

3

liminariesPre1.finiteThenextreflectionstatemengroupstisbut(as[Spr84,remarkPropedositioninloc.3.3(a)],cit.)whicgeneralizeshtheretoisCostatedxeteronlygroups.for
Wegiveapurelycombinatorialproof.
θPropsphericalositionθ-stable1.3.3.subsetLetIwof∈SInvand(Ws1),.b.e.a,sθh∈-twisteSdsuchthatinvolution.Thenthereexistsa
w=s1∙∙∙sh∙wI∙θ(sh)∙∙∙θ(s1),
wherel(w)=l(wI)+2h.
PrLetolof.(w)We>0provandetheassumeclaimthabyttheinductionclaimonholdsl(w)forallbasedθ-tonwistedthetrivinvialolutionscaseww=1withW.
l(w)<l(w).Ifthereexistss∈Swithl(swθ(s))=l(w)−2,thenbyinductionthere
isnothingtoshow.ByLemma1.3.2,itremainstodealwiththecasethatforalls∈S
withCorollaryl(sw)<2.18]l(wthe)thesetIiden:={tifys∈swSθ|(sl)(sw=)w<l(holds.w)}isByspher[AB08,icalPropandeacositionhreduced2.17andI-
wordcanoccursasaninitialsubwordofareduceddecompositionofw;inparticular,
l(wIw)=l(w)−(wI).Henceifthereexistss∈Ssuchthatl(wIws)<l(wIw)then
l(ws)<l(w).InthiscaseLemma1.3.2impliesl(θ(s)w)<l(w),thusθ(s)∈I.But
thenl(θ(s)ws)=l(θ(s)wIwIws)≤l(θ(s)wI)+l(wIws)=(l(wI)−1)+(l(wIw)−1)=l(w)−2,
contrarytoourhypothesisthatswθ(s)=wholds.Accordinglyforalls∈Swe
havel(wI−w1s)>l(wIw).ThereforewIw=1Wandw=wI.Finally,theobservation
θ(w)=w=wimpliesθ(I)=I.
Remark1.3.4.In[Ric82],Richardsongivesacompletecharacterizationofinvo-
lutionsofCoxetergroups,basedonworkdonein[Deo82]and[How80].Seealso
[Hum90,Section8.2]forabriefsummary.However,weshallnotmakeuseofthisin
ork.wtpresenthe1.4.Chambersystems
ChambersystemswhereintroducedbyTitsin[Tit81].Seealso[AB08,Section5.2],
ei03].[W[Ron89],[BC],Definition1.4.1.LetIbeaset.AchambersystemoverIisapair(C,(∼i)i∈I),
i∈whereI,C∼iisisaannonemptequivyalencesetwhoserelationelemenonthetsasetreofcalledchamcbhamersbsucershandthatifwherec∼ifordeacandh
c∼jdtheneitheri=jorc=d.
Definition1.4.2.TherankofachambersystemoftypeIisthecardinalityofI.
Allchambersystems(andbuildings)consideredinthepresentworkareassumed
tobeoffiniterank.
4

1.4.Chambersystems

Definition1.4.3.Giveni∈Iandc,d∈C,thenciscalledi-adjacenttodifc∼id.
Thechambersc,darecalledadjacentiftheyarei-adjacentforsomei∈I.
FortherestofthissectionletC=(C,(∼i)i∈I)beachambersystemoverI.
Definition1.4.4.AgalleryinCisafinitesequence(c0,c1,...,ck)suchthatcµ∈C
forall0≤µ≤kandsuchthatcµ−1isadjacenttocµforall1≤µ≤k.The
numberkiscalledthelengthofthegallery.GivenagalleryG=(c0,c1,...,ck),
weputα(G)=c0andω(G)=ck.IfGisagalleryandifc,d∈Csuchthat
c=α(G),d=ω(G),thenwesaythatGisagalleryfromctodorGjoinsc
.dandDefinition1.4.5.ThechambersystemCissaidtobeconnectedifforanytwo
chambersthereexistsagalleryjoiningthem.
Definition1.4.6.AgalleryGiscalledclosedifα(G)=ω(G).AgalleryG=
(c0,c1,...,ck)iscalledsimpleifcµ−1=cµforall1≤µ≤k.
GivenagalleryG=(c0,c1,...,ck),G−1denotesthegallery(ck,ck−1,...,c0).Fur-
thermoreifH=(c0,c1,...,cl)isagallerysuchthatω(G)=α(H),thenGHdenotes
thegallery(c0,c1,...,ck=c0,c1,...,cl).
Definition1.4.7.LetJbeasubsetofI.AJ-galleryisagalleryG=(c0,c1,...,ck)
suchthatforeach1≤µ≤kthereexistsanindexj∈Jwithcµ−1∼jcµ.
Definition1.4.8.Giventwochambersc,d,wesaythatcisJ-equivalenttod,if
thereexistsaJ-galleryjoiningcandd;wewritec∼Jdinthiscase.
Notethatsince∼iisanequivalencerelation,canddarei-adjacentifandonlyif
theyare{i}-equivalent.
Definition1.4.9.GivenachambercandasubsetJofI,thesetRJ(c):={d∈C|
c∼Jd}iscalledtheJ-residueofc.IfJ={i},thenRJ(c)iscalledthei-panelof
c(orthei-panelcontainingc);apanelisani-panelforsomei∈I.
Notethat(RJ(c),(∼j)j∈J)isaconnectedchambersystemoverJ.
Definition1.4.10.AchambersystemCoverIiscalledresiduallyconnected
ifthefollowingholds:ForeverysubsetJofI,andforeveryfamilyofresidues
(RI\{j})j∈Jwiththepropertythatanytwooftheseresiduesintersectnontrivially,
wehavethat∩j∈JRI\{j}isan(I\J)-residue.
Lemma1.4.11(Lemma3.6.10in[BC]).LetCbeaconnectedchambersystemover
I.ThenCisresiduallyconnectedifandonlyifthefollowingholds:IfJ,K,Lare
subsetsofIandRJ,RK,RLareJ-,K-,L-residueswhichhavepairwisenonempty
intersection,thenRJ∩RK∩RLisa(J∩K∩L)-residue.

5

liminariesPre1.

Example1.4.12.LetFbeafield,andVan(n+1)-dimensionalvectorspace
ononverFtrivial.vDenoteectorbyP(subspacesV)theofVpro.LetjectivCebespacetheosetverofV.allItmaximalconsistsofflagsallinPprop(Ver),
i.e.,necessarilystrictlydim(Viascending)=iforallsequencesi∈{V11,..<.,Vn2}.<W..e.call<VsucnhofaelemenmaximaltsofflagP(aV).chambThener.
twoIndeed,chamwbeersget(V1the,...,Vstructuren)andof(aU1c,..ham.,bUner)tosystembeiover-adjacenI=tif{1,.and..,nonly}bifyVjdefi=ningUj
foralljdifferentfromi.

Buildings1.5.Inthepresentwork,weareonlyinterestedin(twin)buildingscomingfromagroup
witha(twin)BN-pair,i.e.,(twin)buildingsadmittingastronglytransitivegroup
action.Ourmainreferencefor(twin)buildingsis[AB08].Fordetailedtreatmentsof
onthetwintheoryofbuildings,buildings,seealsowe[Titalso92],referto[Rém02],[Bro89],[Ron02],[Ron89],[Müh02].[Tit74],[Wei03].Formore
Definition1.5.1.Let(W,S)beaCoxetersystem.Abuildingoftype(W,S)isa
pair(C,δ)whereCisanonemptysetandδ:C×C→Wisadistancefunction
satisfyingthefollowingaxioms,wherex,y∈Candw=δ(x,y):
(Bu1)w=1ifandonlyifx=y;
(Bu2)ifz∈Cissuchthatδ(y,z)=s∈S,thenδ(x,z)∈{w,ws},andiffurther-
morel(ws)=l(w)+1thenδ(x,z)=ws;
(Bu3)ifs∈S,thereexistsz∈Csuchthatδ(y,z)=sandδ(x,z)=ws.
Forabuilding(C,δ)oftype(W,S)ands∈S,wedefinearelation∼,where
c,d∈Cares-equivalent,i.e.,c∼sd,ifandonlyifδ(c,d)∈{1W,s}.Fromthesaxioms
cabhamoveberitfollosystemws(seethat[AB08,thisisSeinctionfactan5.1.1]).equivOnealencecanactuallyrelation,andcompletely(C,(∼s)s∈Sreconstruct)isa
thebuildinganditsdistancefunctionfromthischambersystem.Hence,inthe
folloparticular,wing,wweewillwillnotspeakofdistinguishgallebries,etweeresiduesntheandbuildingpanelsandofaitscbuilding.hambersystem.In
Definition1.5.2.Therankofabuildingoftype(W,S)is|S|.
atAleastbuildingthreeis(resp.thickexactly(resp.twthino))cifhamforbersanyss∈-adjacenSandttoancy.chamberc∈Cthereare
Example1.5.3.Let(W,S)beaCoxetersystem.DefineδS:W×W→W:
((Wx,,yS)).→Itx−is1y.notThenhardδtoSisseeathatdistanceanythinfunctionbuildingand(ofWt,ypδSe)(isW,aS)thinisbuildingisometricoftotypthise
one.

6

winT1.6.Buildings

Inthepresenttext,allbuildingsareassumedtobeoffiniterankandthick.
Foranytwochambersxandywedefinetheirnumericaldistancel(x,y)asl(δ(x,y)).
Definition1.5.4.Suppose(C,δ)isabuildingoftype(W,S).Thenanapartment
ofCisasubsetΣofC,suchthat(Σ,δ|Σ)isisometricto(W,δS)(cf.Example1.5.3).
Definition1.5.5.AbuildingiscalledsphericalifitsCoxetersystem(W,S)is
ifδ(sphericalc,d)=(i.e.,wS,thefinite).longestInaelemensphericaltof(Wbuilding,,S).twochambersc,darecalledopposite
Definition1.5.6(Cf.Definition5.35from[AB08]).LetRbearesidueofC.
(1)Givend∈C,theuniquechamberc∈Ratminimaldistancefromdiscalled
theprojectionofdontoRandisdenotedbyprojRd.
(2)IfSisanotherresidue,wesetprojRS:={projRd|d∈S}andcallitthe
projectionofSontoR.ThusprojRSisasubsetofR.
NotethatprojRSisactuallyaresidueonitsown(cf.Lemma5.36inloc.cit.).
Definition1.5.7.AnonemptysubsetM⊂Ciscalledconnectedifforanytwo
cinhamM.bersMoreoc,dv∈er,MM,istherecalledisaconvgalleryexifbetforwaneenytcwoandcdhambwhicershc,isd∈Mcompletely,everyconminimaltained
galleryjoiningcanddinCiscontainedinM.
Forexample,Cisconnectedandconvex;andsoiseveryresidue.Also,theinter-
sectionofafamilyofconvexsetsisconvex.
Example1.5.8.ThechambersystemofaprojectivespaceP(V)asdefinedinEx-
ample1.4.12actuallyisabuilding,withCoxetergroupSymn+1ifVis(n+1)-
dimensional.Seee.g.[AB08,Section4.3]fordetails.

BuildingswinT1.6.Twinbuildingsgeneralizesphericalbuildingsinthesensethatthereisstilltheno-
ctionhamofberstwoinchaonembofersthebteingwooppbuildingsosite,onlymaybthatenooppwositetwotobuildingscertaincarehaminvbolversed,inandthe
otherbuilding,andviceversa.Thisismadeprecisebythefollowingaxiomsand
consequences.theirDefinition1.6.1.Atwinbuildingoftype(W,S)isatriple(C+,C−,δ∗)consisting
oftwobuildings(C+,δ+)and(C−,δ−)oftype(W,S)togetherwithacodistance
functionδ∗:(C+×C−)∪(C−×C+)→W
satisfyingthefollowingaxioms,whereε∈{+,−},x∈Cε,y∈C−εandw=δ∗(x,y):
(Tw1)δ∗(y,x)=w−1;

7

liminariesPre1.

(Tw2)if∗z∈C−εissuchthatδ−ε(y,z)=s∈Sandl(ws)=l(w)−1,then
δ(x,z)=ws;
(Tw3)ifs∈S,thereexistsz∈C−εsuchthatδ−ε(y,z)=sandδ∗(x,z)=ws.
Weremindthereaderthatinthisthesis,allbuildingsarethickandoffiniterank.
Fortherestofthissectionlet(C+,C−,δ∗)beatwinbuildingoftype(W,S),and
ε∈{+,−}.Forx∈Cεandy∈C−εweputl∗(x,y)=l(δ∗(x,y)).
Inviewof(Tw1),theothertwoaxiomshavethefollowing“left”analogues:
(Tw2’)ifz∈Cεissuchthatδε(x,z)=s∈Sandl(sw)=l(w)−1,thenδ∗(z,y)=
;sw(Tw3’)ifs∈S,thereexistsz∈Cεsuchthatδε(x,z)=sandδ∗(z,y)=sw.
Asexplainedintheprevioussection,thebuildingsCεmaybeviewedaschamber
systemsoverS.
Definition1.6.2.Aresidue/panel/galleryinCisaresidue/panel/gallery
ineitherC+orC−.
Definition1.6.3.Wesaythattwochambersc∈Cεandd∈C−ε(ε∈{+,−})are
opposite,andwritecoppd,ifδ∗(c,d)=1W.TworesiduesRinC+andSinC−are
calledoppositeiftheyhavethesametypeandcontainoppositechambers.
Definition1.6.4.AtwinapartmentofatwinbuildingCisapairΣ=(Σ+,Σ−)
suchthatΣ+isanapartmentofC+,Σ−isanapartmentofC−,andeverychamber
inΣ+∪Σ−isoppositepreciselyoneotherchamberinΣ+∪Σ−.
Thereisageneralizationofthenotionofprojectionsfrombuildingstotwinbuild-
ings,atleastforsphericalresidues:
Lemma1.6.5(E.g.Lemma5.149from[AB08]).IfRisaresidueinCεofspherical
type,anddisachamberinC−ε,thenthereisauniquechamberc∈Rsuchthat
δ∗(c,d)isofmaximallengthinδ∗(R,d).Thischambersatisfies
δ∗(c,d)=δε(c,c)δ∗(c,d)
forallc∈R.WecallctheprojectionofdontoRanddenoteitbyprojR(d).
Usingthisextendednotionofprojections,wecanalsogeneralizetheconceptof
convexity.
Definition1.6.6(Cf.Definition5.158from[AB08]).Apair(M+,M−)ofnonempty
subsetsM+⊆C+andM−⊆C−iscalledconvexifprojPc∈M+∪M−forany
c∈M+∪M−andanypanelP⊆C+∪C−thatmeetsM+∪M−.
Remark1.6.7.Anequivalentwayofdefiningconvexityisthefollowing:Apair
(M+,M−)ofnonemptysubsetsofM+⊆C+andM−⊆C−isconvexifandonlyifit
jections.prounderclosedis

8

-pairsNB1.7.

Examplemorphism1.6.8.unique)AntywinningsphericalwithabuildingcopyCC+−ofoftypeitself(W,(seeS)[Tit92,admitsanProp(upositiontoiso1]-
orthee.g.copy[AB08,(the“twin”)Exampleofc5.136]inCforbycdetails):.TheForandistanceychamonbCerc+then∈isC+,defineddenoteas
δ(c,d):=wδ(c,d)+w,and−the−codistancebetweenthe−twobuildingsvia
δ−∗(c+−,d−−)=δ+(0c++,d++)w0+and0δ∗(d−,c+)=w0δ+(d+,c+),wherew0isthelongest
.WoftelemenducedInthisforbuildingconstruction,sandtheoncetwforotwindefinitionsbuildings,ofbeingcoincideopposite,hereinwhicthehwfolloeoncewinginsense:tro-
Ifoppc+,osited+(∈i.e.Cδ+∗(care,dopp)o=sitδe(c(i.e.,,dδ)+w(c+=,d1+).=w0)ifandonlyifc+andd−are
+−+++0W
Likewise,projectionsinsideC+correspondnaturallytoprojectionsbetweenC+
.and−CdifferenDefinitionthalves1.6.9.oftheTwobuilding)residuesareRandcalledQparallel(assumedifprotojbe(Q)sphe=Rricalandifprotheyj(areR)=in
QR.Q

-pairsNB1.7.Ourmainreferenceforthissectionis[AB08,Section6.2],whereallclaimsmade
belowareproved.Anotherexcellentreferenceis[Tit74].Finally,[Bou68,Chapter
IV](anditsEnglishtranslation[Bou02])seemtocontaintheoriginaldefinition.
Definition1.7.1.WecallapairofsubgroupsBandNofagroupGaBN-pairif
BandNgenerateG,theintersectionT:=B∩NisnormalinN,andthequotient
groupW:=N/TadmitsasetSofgeneratorssuchthatthefollowingconditions
hold:(BN1)wBs⊆BwsB∪BwBforallw∈W,s∈S;
(BN2)sBs−1⊂Bforalls∈S.
ThegroupWiscalledtheWeylgroupassociatedtotheBN-pair.Thequadruple
(G,B,N,S)isalsocalledaTitssystem.
Wecollectsomewell-knownfactsaboutagroupGadmittingaBN-pair:
•(W,S)isaCoxetersystem.
•G=w∈WBwB,theBruhatdecompositionofG.
•AnyconjugateofBiscalledaBorelsubgroup.
•ForeachsubsetJ⊂SthesetPJ:=w∈WJBwBisasubgroupofG,called
standardparabolicsubgroupoftypeJ.AnyconjugateofPJiscalleda
.subgroupolicparab

9

liminariesPre1.

•AG/BTitsandsystemwhose(G,Bdistance,N,S)functionleadsδto:aG/B×buildingG/B→whoseWissetgivofencbyhamδ(bgersB,hBequals)=
wifandonlyifBh−1gB=BwB.
Example1.7.2.ThestandardexampletonamehereisthegroupG=SLn(F)
overanyfieldF,withn≥2.LetBbethegroupofuppertriangularmatrices
inG,letNbethegroupofmonomialmatricesinG.Onereadilyverifiesthat
G=B,N.NowT=B∩NisthegroupofdiagonalmatricesinG,andW=N/T
isclearlyisomorphictothegroupgeneratedbyalln×npermutationmatrices,
whenceisomorphictoSymn,thegroupofallpermutationsoftheset{1,...,n}.In
particular,itisaCoxetergroup.
Moreover,GactsnaturallyonthevectorspaceV=Fn,butalsoontheprojective
spaceP(V)(seeExamples1.4.12and1.5.8).Indeed,BisthestabilizerinGofthe
ofallmaximalchamflagbers.e1Th<us,e1,eone2<.obtains..<ae1,..bijection.,en,betandweenGactsthechamtransitivberselyofonP(Vthe)andset
thecosetspaceG/B,inaccordancewiththefactsweassembledabove.
Example1.7.3.LetGbeaconnectedreductivealgebraicgroupoveranalge-
braicallyclosedfield.TakeanyBorelgroupB,andanymaximaltorusTcontained
inB.LetN:=NG(T).Then(B,N)formaBN-pairinG.
Moregenerally,ifGisaconnectedreductive(possiblynon-split)F-groupforany
offieldaF,minimalthenG(parabF),olictheFgroup-subgroupofFB-rationalandptheointsofnormalizerG,pNossessesofaaBmaximalN-pairF-splitconsistingtorus
.BintainedconRemark1.7.4.Theprecedingexampleindicatesthatourchoiceofcallingthecon-
jugatesofthesubgroupB“Borelsubgroup”issomewhatunfortunate,asitisincon-
sistentwiththetheoryofalgebraicgroups.In[AB08],theauthorsusethetermTits
subgroupinstead,whichavoidsthisconfusion.However,forthepresentthesis,we
stickwiththetermBorelsubgroupasitseemstobemorecommonintheliterature.

1.8.TwinBN-pairs
Referencesinclude[Tit92,Section3.2]and[AB08,Section6.3.3].
Definition1.8.1.Let(G,B+,N,S)and(G,B−,N,S)betwoTitssystemssuch
thatB+∩N=B−∩N,i.e.,withequalWeylgroups.Then(B+,B−,N)iscalleda
twinBN-pairwithWeylgroupWifthefollowingconditionsaresatisfied:
(TBN1)BεwB−εsB−ε=BεwsB−εforε∈{+,−}andallw∈W,s∈Ssuchthat
l(ws)<l(w);
(TBN2)B+s∩B−=∅foralls∈S.
Inthiscase,wealsosaythatthetuple(G,B+,B−,N,S)isatwinTitssystem.A
twinBN-pairiscalledsaturatedifB+∩B−=T.

10

systemsgroupotRo1.9.

Example1.8.2(E.g.Section6.5in[AB08]).ContinuingtheSLn(F)-examplefrom
above,wegetatwinBN-pairinSLnbytakingasB+resp.B−theupperresp.lower
triangularmatrices,andforNthemonomialmatrices.Again,T=B+∩N=B−∩N
consistsofthediagonalmatrices.SinceT=B+∩B−,thisisinfactasaturatedtwin
-pair.NBBesidestheBruhatdecompositionswithrespecttobothB+andB−(aconsequence
ofthefactthat(B+,N)and(B−,N)areBN-pairs),agroupGwithatwinBN-pair
admitstheso-calledBirkhoffdecomposition
G=BεwB−ε,whereε∈{+,−}.
W∈wDefinition1.8.3.Forε∈{+,−},anyconjugateofBεiscalledaBorelsubgroup
ofsignε.ForeachsubsetJ⊂SthesetP:=w∈WJBεwBεisasubgroupofG,
calledstandardparabolicsubgroupoftypeJandsignε.AnyconjugateofPJ
iscalledparabolicsubgroupofsignε.
Remark1.8.4.AgroupGwithatwinBN-pairhenceyieldstwobuildingsG/B+
andG/B−withdistancefunctionsδ+andδ−.Furthermore,usingtheBirkhoff
decompositionwecandefinethecodistancefunctionδ∗:(G/B−×G/B+)∪(G/B+×
G/B−)→Wviaδ∗(gB−,hB+)=wifandonlyifB+h−1gB−=B+wB−and
δ∗(hB+,gB−):=(δ∗(gB−,hB+))−1.Thetuple((G/B+,δ+),(G/B−,δ−),δ∗)thenis
atwinbuilding,thetwinbuildingassociatedtoG.
Example1.8.5(E.g.Sections6.9and6.12in[AB08]).Letn≥2.Abovewehave
seenthatthegroupSLn(F)overanyfieldFadmitsanaturaltwinBN-pair.However,
intheaboveexample,B+andB−areconjugate,andeachhalfofthebuildingis
spherical.Thismeansthatwedonotreallygetanythingnewfromthetwinning.
NowconsiderSLn(F[t,t−1]),whereFisanarbitraryfieldandF[t,t−1]istheringof
LaurentpolynomialsoverF.Again,wecanendowthiswiththeBN-pairconsisting
ofthegroupsofupperandlowertriangularmatrices,aswellasNequaltothegroups
ofmonomialmatrices.Butthereisasecond,fundamentallydifferenttwinBN-pair:
LetB+bethesetofmatricesinSLn(F[t])whichareuppertriangularmodulot,
andlikewiseletB−bethesetofmatricesinSLn(F[t−1])whichareuppertriangular
modulot−1.Finally,Nasbeforeisthesetofmonomialmatrices.
Onecannowverifythat(B+,B−,N)indeedconstitutesatwinBN-pair.More-
over,usingbasicmatrixcalculations,onecanreadilyverifythatB+andB−arenot
conjugateinsideG.Indeed,theWeylgroupofthistwinBN-pairisoftypeAn−1,in
particularinfinite.Sothetwoassociatedbuildingsarenotspherical,andwegeta
building.wint“true”

“true”twinbuilding.

systemsgroupotRo1.9.

Thefollowingdefinitionisbasedon[AB08,Definition7.82andSection8.6.1],which
inturnisderivedfrom[Tit92,Section3.3].Seealso[CR08]foranotheraccessible
duction.troin

11

liminariesPre1.

Definition1.9.1.LetGbeagroupendowedwithafamily{Uα}α∈Φofsubgroups,
indexedbyarootsystemΦoftype(W,S).LetTbeanothersubgroupofG.Then
thetriple(G,{Uα}α∈Φ,T)iscalledanRGD-systemoftype(W,S)ifitsatisfies
theconditionsbelow,whereU±:=Uα|α∈Φ±:
(RGD0)Foreachα∈Φ,wehaveUα={1}.
(RGD1)Foreveryprenilpotentpair{α,β}⊂Φofdistinctroots,wehave[Uα,Uβ]⊂
Uγ|γ∈]α,β[.
(RGD2)Foreachs∈Sandeachu∈Uαs\{1},thereexistelementsu,uofU−αs
suchthattheproductµ(u):=uuuconjugatesUβontoUs(β)foreach
.Φβ∈(RGD3)Foreachs∈SwehaveU−αs⊆U+.
(RGD4)G=T.Uα|α∈Φ.
(RGD5)TnormalizesUαforeachα∈Φ,i.e.,
T≤NG(Uα).
Φ∈αThentheUαarecalledrootsubgroupsandthepair({Uα}α∈Φ,T)isreferredtoas
arootgroupdatum.
Westatethefollowingwithoutproof,butreferthereaderto[Tit92,Proposition
4]or[AB08,Theorem8.80]fordetails.
Proposition1.9.2.Let(G,{Uα}α∈Φ,T)beanRGD-systemoftype(W,S).Define
N:=T.µ(u)|u∈Uα\{1},α∈Π,
B+:=T.U+,
B−:=T.U−.
Then(G,B+,B−,N,S)isasaturatedtwinBN-pairofGwithWeylgroupN/T=∼W.
WecallitthetwinBN-pairassociatedtotherootgroupdatum.
Hence,toeveryRGD-system,a(Moufang)twinbuildingisassociatedinanatural
.yawDefinition1.9.3.AnRGD-system(G,{Uα}α∈Φ,T)iscalledfaithfulifGoperates
faithfullyontheassociatedbuilding.ItiscalledcenteredifGisgeneratedbyits
rootgroups,andreducedifitisbothcenteredandfaithful.
Lemma1.9.4(E.g.Lemma8.55andSection8.8in[AB08]).NG(U+)=B+=
NG(B+)andNG(U−)=B−=NG(B−).

12

1.10.MoufangsetsandpointedMoufangsets

AsaconsequenceofthisLemma,thefollowingiswell-defined:
Definition1.9.5.LetB=gBεg−1beanarbitraryBorelsubgroup,whereε∈
{+,−}.ThentheunipotentradicalU(B)ofBisthecorrespondingconjugate
B=gU±g−1ofU+orU−.
Remark1.9.6.Ingeneral,thegroupU(B)willbeneithernilpotentnoraradical,so
thenameunipotentradicalissomewhatmisleading.Nevertheless,wechosethisname
inlackofabetterone,andsinceitisalsousedlikethatelsewhereintheliterature,
e.g.[CM06].Notealsothatonecandefineunipotentradicalsgeometricallyandfor
arbitraryparabolicsubgroupsofsphericaltype,butwedonotneedthishere.
Definition1.9.7(SeeSection3.3in[Tit92]).ForanyRGD-system(G,{Uα}α∈Φ,T),
denotebyG◦thequotientofthesubgroupUα|α∈Φbyitscenter,andbyUα◦the
canonicalimageofUαinG◦.Unlessthere◦existsarootorthogonalto◦allother◦roots,
athereducedcanonicalRGD-systemhomomorphismswiththeUαsame→Uαassoareciatedtisomorphisms.winbuildingThenas((GG,,{{UUαα}}α∈α∈Φ,Φ)T)is.
Forthisreason(G◦,{Uα◦}α∈Φ)iscalledthereductionof(G,{Uα}α∈Φ,T).
Definition1.9.8.WesetXα:=Uα,U−αandXα,β:=Xα,Xβ.Arootgroup
datumiscalledlocallysplitifthegroupTisabelianandifforeachα∈Φthere
isafieldFαsuchthatXαisisomorphictoSL2(Fα)orPSL2(Fα)and{Uα,U−α}is
isomorphictoitsnaturalrootgroupdatum.Alocallysplitrootgroupdatumis
calledF-locallysplitifFα=Fforallα∈Φ.
Connectedreductivealgebraicgroupsand(split)Kac-Moodygroupsareexamples
ofgroupswitharootgroupdatum,cf.Chapter6.

1.10.MoufangsetsandpointedMoufangsets
anInthisadaptionsectionofwegiv[DMGH09,eabriefinSectiontro5].ductioFornatomoreMoufangcompletesets.inThetrotextductioninthistosectionMoufangis
[DMS].e.g.seesets,InordertobeconsistentwiththestandardnotationusedinthetheoryofMoufang
setswewillalwaysdenotetheactionofapermutationonasetontheright,i.e.,we
willwriteaϕratherthanϕ(a).
Definition1.10.1.AMoufangsetisasetXofsizeatleasttwotogetherwith
aandcollectionactingofregularlygroups((i.e.,Ux)x∈Xsharply,suchthattransitiveacehly)UxonisXa\{x}subgroup,andofsucSymh(thatX)eacfixinghUxx
pcalledermutesthethelittleset{proUy|jectivy∈eX}groupbyoftheconjugation.MoufangTheset;grouptheG:=groupsUxUx|xare∈Xcalledis
.groupsotro

13

liminariesPre1.

OurapproachtoMoufangsetsistakenfrom[DMW06].LetM=(X,(Ux)x∈X)
beanarbitraryMoufangset,andassumethattwooftheelementsofXarecalled0
and∞.LetU:=X\{∞}.Eachα∈U∞isuniquelydeterminedbytheimageof0
underα.If0α=a,wewriteα=:αa.HenceU∞={αa|a∈U}.WemakeUinto
a(notnecessarilyabelian)groupwithcomposition+andidentity0,bysetting
a+b:=aαb.(1.1)
Clearly,U=∼U∞.NowletτbeanelementofGinterchanging0and∞.(Suchan
elementalwaysexists,sinceGisdoublytransitiveonX.)Bythedefinitionofa
Moufangset,wehave
U0=Uτ∞andUa=U0αa
foralla∈U.Inparticular,theMoufangsetMiscompletelydeterminedbythe
groupUandthepermutationτ;wewilldenoteitbyM=M(U,τ).
Remark1.10.2.Inviewofequation(1.1),itmakessensetousetheconvention
thata+∞=∞+a=∞foralla∈U.
Definition1.10.3.Foreacha∈U,wedefineγa:=αaτ,i.e.,xγa=(xτ−1+a)τfor
allx∈X.Consequently,U0={γa|a∈U}.
Definition1.10.4.Foreacha∈U∗=U\{0},wedefineaHuamaptobe
ha:=ταaτ−1α−(aτ−1)τα−(−(aτ−1))τ∈Sym(X);
ifweusetheconventionofRemark1.10.2,thenwecanwritethisexplicitlyas
ha:X→X:x→(xτ+a)τ−1−aτ−1τ−−(aτ−1)τ.WedefinetheHua
subgroupofMasH:=ha|a∈U∗.
Remark1.10.5.Observethateachhafixestheelements0and∞.By[DMW06,
Theorem3.1],thegroupHequalsG0,∞:=StabG(0,∞),andby[DMW06,Theo-
rem3.2],therestrictionofeachHuamaptoUisadditive,i.e.,H≤Aut(U).
Definition1.10.6.Foreacha∈U∗,wedefineaµ-mapµa:=τ−1ha.
NotethatµaistheuniqueelementinthesetU0∗αaU0∗interchanging0and∞.In
particular,µa−1=µ−a.
Definition1.10.7.Let(X,(Ux)x∈X)and(Y,(Vy)y∈Y)betwoMoufangsets.A
bijectionβfromXtoYiscalledanisomorphismofMoufangsets,iftheinduced
mapχβ:Sym(X)→Sym(Y):g→β−1gβmapseachrootgroupUxisomorphically
ontothecorrespondingrootgroupVxβ.AnautomorphismofM=(X,(Ux)x∈X)
isanisomorphismfromMtoitself.ThegroupofallautomorphismsofMwillbe
denotedbyAut(M).
NowweintroducepointedMoufangsets,whichwillbeMoufangsetswithafixed
identityelement.Wewillthen,inanalogywiththetheoryofJordanalgebras,
introducethenotionsofanisotopeofapointedMoufangset,andwewilldefine
JordanisomorphismsbetweenMoufangsets.

14

1.10.MoufangsetsandpointedMoufangsets

Definition1.10.8.ApointedMoufangsetisapair(M,e),whereM=M(U,ρ)
isaMoufangsetandeisanarbitraryelementofU∗.Theτ-mapofthispointed
Moufangsetisτ:=µ−e=µe−1,andtheHuamapsarethemapsha:=τµa=µ−eµa
foralla∈U∗.WealsodefinetheoppositeHuamapsga:=τ−1µa=µeµaforall
a∈U∗.Clearly,M=M(U,τ)=M(U,τ−1).
Definition1.10.9.Let(M,e)and(M,e)betwopointedMoufangsets,withM=
M(U,ρ)andM=M(U,ρ).Apointedisomorphismfrom(M,e)to(M,e)
isanisomorphismfromUtoUmappingetoeandextendingtoaMoufangset
isomorphismfromMtoM(bymapping∞to∞).Apointedisomorphismfrom
(M,e)toitselfiscalledapointedautomorphismof(M,e),andthegroupofallpointed
automorphismsisdenotedbyAut(M,e).
Definition1.10.10.Let(M,e)beapointedMoufangset,andleta∈U∗bearbi-
trary.Then(M,a)iscalledthea-isotopeof(M,e),orsimplyanisotopeifone
doesnotwanttospecifytheelementa.Theτ-mapandtheHuamapsof(M,a)will
bedenotedbyτ(a)andhb(a),respectively.Observethat
τ(a)=µ−aandhb(a)=µ−aµb=ha−1hb
foralla,b∈U∗.
Remark1.10.11.Ournotionofana-isotopeis,inacertainsense,theinverseof
theusualnotionofana-isotopein(quadratic)Jordanalgebras,whereoura-isotope
wouldbecalledthea−1-isotope(wherea−1denotestheinverseintheJordanalgebra)
andwherehb(a):=hahb.Itis,inthegeneralcontextofMoufangsets,notnatural
totrytobecompatiblewiththisconvention,becauseha−1isingeneralnotofthe
formhbforsomeb∈U∗.Infact,wehaveha−1=gaτforalla∈U∗;see[DMW06,
3.8(i)].LemmaDefinition1.10.12.Let(M,e)and(M,f)betwopointedMoufangsetswith
M=M(U,ρ)andM=M(U,ρ),andwithHuamapshaandha,respectively.An
isomorphismϕfromUtoUiscalledaJordanisomorphismif(bha)ϕ=(bϕ)haϕ
foralla,b∈U∗.AJordanisomorphismfrom(M,e)to(M,a)iscalledanisotopy
from(M,e)toitsa-isotope.Explicitly,amapϕ∈Aut(U)isanisotopyifandonly
ifhaϕ=ϕh(aϕeϕ)
foralla∈U∗.Thegroupofallisotopiesfrom(M,e)toanisotopeiscalledthe
structuregroupof(M,e),andisdenotedbyStr(M,e).
NotethatitisnotclearwhetherStr(M,e)≤Aut(M).AlsoobservethatG∩
Str(M,e)=H;wecallHtheinnerstructuregroupof(M,e).

15

1.

16

liminariesPre

CHAPTEROTW

FLIPS

InSection2.1,weintroducetheconceptofflips(andtheirslightlymoregeneral
siblings,thequasi-flips),firstinthecontextoftwinbuildings,theninthecontextof
ingroupsterchangewithtthewintwBoNhalv-pairs.esofaFlipstwinareessenbuilding,tiallyresp.invtheolutoryconjugacyautomorphismsclassesofwhicBorelh
groupsofplusandofminussign.
Wedemonstratetheclosecorrespondencebetweenthesetwokindsofflips(on
buildingsandongroups)inSection2.2,whereweprovethatatwinBN-quasi-flipof
abuildinggroupGassowithciatedtwintoBGN.-pairTheconvinduceserseaisshouniquewnttowinholdbuildingundercertainquasi-fliponconditionsthetwinas
ell.wpAortanbrieftspecialdetourclassinofSectionflips,for2.4iswhichusedatolotinoftrotheducetheorystrongdevflipselop,edwhichthroughoutareanthisim-
thesissimplifiesconsiderably,permittingmoreuniformandmaybealsomoreele-
gantapproaches.Also,flipsthatwerestudiedinthepast(e.g.aspartofthePhan
program),haveusuallybeenstrong.
yetInimpSectionortant2.3toolweinthenthebrieflyfurtherpresenstudytoftheflipstnotionhroughoutofsteepthedescrestentof.thisThiscisahapterbasicas
wellasinlaterpartsofthepresentthesis.
Infact,inthefollowingSection2.5,weapplysteepdescenttoprovethatunder
somemildconditions,anychamberofatwinbuildingwithaquasi-flipθiscontained
inaθ-stableapartment.InSection2.6westudyinsomemoredetailwhenthe
aforementionedmildconditionsaresatisfied.
presenThisted:allAthendoubleculminatescosetindecompSectionosition2.7,ofawheregrouptheGmainendowedresultwithofathisRcGD-systemhapteris
andaquasi-flipθ,generalizingpreviousresultsonalgebraicgroups(incharacteristic
differentfrom2)andKac-Moodygroups(incharacteristic0,foralgebraicallyclosed
fields).

17

Flips2.

2.1.BuildingflipsandBN-flips
Throughoutthissection,C=(C+,C−,δ∗)denotesaMoufangtwinbuildingoftype
(W,S)(seeSection1.6).Moreover,GisagroupactingstronglytransitivelyonC,
henceisendowedwithatwinBN-pair(B+,B−,N)(seeSection1.8).

Buildingquasi-flipsWenowpresentthedefinitionofabuildingflip,aconceptwhichhasbeenintroduced
in[BGHS03],albeitinadifferentform.Herewegiveamoregeneraldefinition
comparedtowhatappearedpreviouslyintheliterature,subsumingallkindsof
buildingflipsknowntous.
Definition2.1.1.Abuildingquasi-flipofCisapermutationθofC+∪C−with
thefollowingproperties:
(1)θ2=id;
(2)θ(C+)=C−;
(3)θpreservesadjacencyandopposition,i.e.,forε∈{+,−}andforallx,y∈Cε,
z∈C−εwehavex∼yifandonlyifθ(x)∼θ(y);andxoppzifandonlyif
θ(x)oppθ(z).
,additionallyIf,(3*)θflipsthedistancesandpreservesthecodistance,i.e.,forε∈{+,−}and
forallx,y∈Cε,z∈C−εwehaveδε(x,y)=δ−ε(θ(x),θ(y));andδ∗(x,z)=
δ∗(θ(x),θ(z)),
wecallθabuildingflip.
Remark2.1.2.InExample1.6.8,wesawthatanysphericalbuildingC+oftype
(W,S)admitsan(uptoisomorphismunique)twinningwithacopyC−ofitself.
LetφbeanarbitraryinvolutoryautomorphismofC.Thenweobtainaquasi-
flipθasfollows:Forc+∈C+,supposed+=φ(c+).Thendefineθ(c+):=d−and
θ(c−):=d+.Thisisawell-definedmapoforder2,interchangingthehalvesofthe
sphericaltwinbuilding,preservingadjacencybyvirtueofitsdefinition.Duetothe
wayδ∗wasdefined,thisalsoimpliesthatθpreservesopposition.
Moregenerally,ifφisanalmostisometryinthesenseof[AB08,Section5.5.1](that
is,itisanisometryuptoapermutationofS),wecanderiveabuildingquasi-flip
fromitassketchedabove.
Conversely,anyquasi-flipθofasphericaltwinbuildinginducesanalmostisometry
φonthepositivehalfofthetwinbuildingbysettingφ(c+):=θ(c−).Thiswillbea
consequenceofLemma2.1.4below.

18

2.1.BuildingflipsandBN-flips

Inviewofthisremark,wemayoccasionallytalkaboutquasi-flipsofspherical
buildings,whichbytheabovearesimplyalmostisometriesoforder2.
Historically,thefollowing(spherical)exampleistheprototypeofallflips,see
[BS04].Example2.1.3.LetVbean(n+1)-dimensionalvectorspaceoverafieldFofchar-
acteristicdifferentfrom2.DenotebyCthesphericalbuildingoftypeAnassociated
toV,whichisthechambercomplexoftheprojectivespaceP(V)ofpropernontrivial
subspacesofV.(Seee.g.[AB08,Section4.3]fordetails.)
Onthisspace,consideranon-degenerateunitaryororthogonalform.Fromthis
weobtainaninvolutorymapφonthechambersystemasfollows:Recallthata
chamberisamaximalflagofsubspacesc=(V1<V2<...<Vn),wheredimVi=i.
WesendeachVitoitsorthogonalcomplementVi⊥ofdimensionn+1−i.Thuscis
mappedtothechamber(Vn⊥<...<V1⊥).Theresultisaso-calledpolarity.
Toseethatthisisactuallya(quasi-)flipinoursense,firstrecallRemark2.1.2,
whichsaysthatanyalmostisometryofasphericalbuildinginducesaquasi-flipof
thecorrespondingtwinbuilding.Now,weclaimthatproperty(3*)holds,whichat
firstmightseemcounter-intuitive,aswemapi-dimensionalsubspacesto(n−i+1)-
dimensionalsubspaces,whichisnottypepreservinginthesphericalsetting,but
ratherinducesadiagramautomorphism.But(3*)wasdefinedinthetwinbuilding
context,sotakingRemark2.1.2intoaccount,wehavetocheckwhetherδ+(c+,d+)=
δ−(φ(c−),φ(d−))=w0δ+(φ(c+),φ(d+))w0.Itturnsoutthatconjugatingbyw0pre-
ciselycancelsthetypechangingeffectofthediagramautomorphism.

Itisclearthatproperty(3*)impliesproperty(3).Theconverseisnottrue,but
holds:wingfollotheLemma2.1.4.Letθbeabuildingquasi-flip.Thenθinducesanautomorphism
θ˜oftheCoxetersystem(W,S)oforderatmost2,suchthatforε∈{+,−}and
forallx,y∈Cε,z∈C−εwehaveθ˜(δε(x,y))=δ−ε(θ(x),θ(y));andθ˜(δ∗(x,z))=
δ∗(θ(x),θ(z)).Inparticular,θ˜permutesS.
Proof.Fixachamberc.Byitsdefinition,θmapspanelstopanels.Henceforevery
s∈Sthereexistst∈S,suchthatthes-panelofcismappedtothet-panelofθ(c)
andviceversa.WeobtainapermutationσofthesetS.Wehavetoprovethat
thispermutationhasorderatmost2andextendsuniquelytoanautomorphismθ˜of
(W,S)satisfyingallclaimedproperties.
Westartbyarguingthattwinapartmentsaremappedtotwinapartments:Let
Σbeanarbitrarytwinapartmentcontainingc.LetdbetheuniquechamberinΣ
oppositec.Fromthedefinitionofθitisclearthatitpreservesnumericaldistances,
andthusconvexsetsofeitherhalfofthetwinbuilding.Toconcludethatitmaps
twinapartmentstotwinapartments,itremainstoshowthatitalsopreservesthe
distance.coumericalnForarbitrarys∈S,letcbetheprojectionofdtoPs(c)anddtheprojection
ofctoPs(d).Thenc,darecontainedinΣandoppositetoeachother.Sinceθ

19

Flips2.

preservesadjacency,thereexistt,r∈Ssuchthatθ(c)∼tθ(c)andθ(d)∼rθ(d).
Additionally,ascisoppositedbutnotopposited,wefindthatθ(c)isopposite
θ(d),butnotoppositeθ(d).Henceby(Tw2)wemusthaveδ∗(θ(c),θ(d))=t.But
δ(θ(d),θ(d))=randδ∗(θ(c),θ(d))=1,thusagainby(Tw2),r=t,orequivalently,
ourpermutationσhasorderatmost2.Inparticular,θ(d)andθ(c)arecontained
inthetwinapartmentspannedbyθ(c)andθ(d).Sincecandswerearbitrary,we
concludethatθ(Σ)isagainatwinapartment.
NowΣandθ(Σ)arebothisomorphictotheCoxetercomplexof(W,S),and
hencetoeachother.Thereisauniquetype-preservingisomorphismιbetweenthem
mappingctoθ(c).Accordingly,θ◦ιisanautomorphismofΣfixingc,whichinduces
awell-definedautomorphismθ˜Σof(W,S)whichcorrespondstothepermutationσ
.StorestrictedwhenButanychamberdiscontainedinatwinapartmentalsocontainingc.Inpar-
ticular,anypanelofthebuildingmeetsatwinapartmentcontainingc.Sinceθ˜is
fullydeterminedbyσ,weconcludethateverys-panelofthebuildingismappedto
aσ(s)-panel.Theclaimfollows.
Inthesequel,wealsodenote,byslightabuseofnotation,theinducedautomor-
phismθ˜ofWbyθ.
Soallinall,thedifferencebetweenabuildingflipandabuildingquasi-flipis
thattheformeristype-preserving,whilethelattermightadditionallyinvolvea
diagramautomorphism.Or,flipsandquasi-flipsarerelatedlikeisometriesand
isometries.almostRemark2.1.5.Therearequasi-flipswhicharenotflips.Forexample,startwithan
AnbuildingasinExample2.1.3,butthistimeletφbeaninvolutoryautomorphism
ofV,say,areflection.Thisisanisometryofthesphericalbuilding,butonthe
twin-building,theinducedquasi-flipisnolongertypepreserving.
Alternatively,takeapolarityψofageneralizedquadrangle.AsinExample2.1.3,
forthistoinduceafliponthetwinbuilding,itwouldhavetosatisfy
δ+(c+,d+)=δ−(ψ(c−),ψ(d−))=w0δ+(ψ(c+),ψ(d+))w0.
Butforaquadrangle,theWeylgroupisoftypeB2=C2,andthelongestelement
w0iscentral,henceψisaflipifandonlyif
δ+(c+,d+)=δ+(ψ(c+),ψ(d+)),
whichwouldbetrueifψwasaninvolutoryautomorphismofthequadrangle,but
sinceitisapolarity,thepropertydoesnothold.

Lateron,wewillbeinterestedin“howfar”aquasi-flipmovesachamber.Thisis
capturedbythefollowingdefinition:

20

2.1.BuildingflipsandBN-flips

Definition2.1.6.Forachamberc,wecallw=δ∗(c,θ(c))∈Wtheθ-codistance
ofcandwriteδθ(c):=w.Wealsosetlθ(c):=l(δθ(c))=l(w),thenumerical
.distance-coθwhicThehfaradmitthestsucahcchamhambbererscanhapvespossiblyecialbepropmapperties,edissotoweangivoppeositethemcahamname.ber.Flips
Definition2.1.7.Wecallabuilding(quasi-)flipproperifthereexistsachamberc
withθ-codistance1W,thatis,δ∗(c,θ(c))=1W.Aproperbuildingflipisalsocalled
.olutionvinPhanGeneralizingtheideaofchambersmappedtooppositeones,wearriveatthe
wing:folloDefinition2.1.8.WecallaresidueRofCaPhanresidueifRisoppositeθ(R)
(meaningthatforeverychamberinRthereexistsachamberoppositetoitinθ(R),
andviceversa).AminimalPhanResidueisaPhanresiduewhichisminimal
byinclusion,i.e.,whichdoesnotcontainanyotherPhanresidue.Finally,wecalla
chambercaPhanchamberifcandθ(c)areopposite.
Withtheaboveterminology,aPhanchamberissimplyachamberwithθ-codistance
1W,anda(quasi)-flipisproperifandonlyifitadmitsaPhanchamber.
Examples2.1.9.AssumeagainthesettingdescribedinExample2.1.3:LetVbe
an(n+1)-dimensionalvectorspaceoverafieldFofcharacteristicdifferentfrom2.
AssociatedtothisistheprojectivespaceP(V)ofpropernontrivialsubspacesofV,
building.sphericala

(1)V.AssumeAsinthatfExampleisa2.1.3,non-degeneratethisinducesunitarya,(quasi-)symplecticflipofPor(V).orthogNoonalw,thisformflipof
isproperifandonly⊥ifthereexists⊥achamber(V1<V2<...<V⊥n)whichis
oppositeitsimage(V<...<V).ThisholdsifandonlyifV∩V={0}
(equivalentlyVi⊕Vn⊥n+1−i=V)for1alli∈{1,...,n}.Thisistrueiifnan+1−yionlyif
theformisanisotropic(i.e.,novectorisorthogonaltoitself).Symplecticforms
areneveranisotropic(there,everyvectorisorthogonalitself),whichleavesthe
forms.orthogonalorunitary(2)ConsiderV=Rn+1,endowedwiththestandardscalarproductwithrespectto
someorthogonalbasise1,...,en+1ofV.Thenthisforminducesaflip,which
weclaimisaproperflip.
Fourorletflipc=sends(e1e<→e1,ee2,.<...,.e.<,e1,ee2,.e..,e→n)eb,e..a.c,ehamb,erandofP(soV).on.TFhenor
12n+1123n+1
ethis3,...,reason,en+1our<e2,.starting..,enc+1ham).berOneisinreadilytercvhangederifieswiththatcd=and(denare+1opp<...osite.<

21

Flips2.

(3)ConsideragainV=R2nbutthistimewithasymplecticfrom(∙,∙)andcor-
respondingbasise1,...,en,f1,...fnwith(ei,fj)=δij,(fi,ej)=−δijand
(ei,ej)=(fi,fj)=0foralli,j∈{1,...,n}.Thenany⊥subspaceUofodd
dimension(e.g.e1)hasanontrivialradical(i.e.,U∩U={0}).SoifV1<
...<V2n−1isamaximalflag,thenV1∩V1⊥=V1,henceV2n−1∩V1⊥=V1={0}.
Accordingly,nochamberismappedtoanoppositeone,andtheflipisimproper.
Infact,theminimalcodistance(maximaldistance)onecanachievebetweena
chamberanditsimageiss1∙s3∙∙∙s2n−1,assumingthediagramA2nislabeled
from1to2n.
Foradetailedanalysisofsymplecticflips,wereferto[BH08].
(4)Continuingthis,therearealsoproperbuildingquasi-flips(nottype-preserving):
aTakequasi-flipthelinearwhichmapisnotsendingaflip,eachandeiittoswen−apsi+1.theWeopphavositeecseenhambthaterscthisanddinduces.
Example2.1.10(See[PT84]).Considernowthepermutationθ=(15)(24)(36)∈
S6.quadrangleClearly,isθaisabuildingnontrivialininwhicvholutiontheandlongestpreservelemenestofdistances.theWeylSincegroupaiscengeneralizedtral,
italsoautomaticallypreservesthecodistance.Henceθisabuildingflip.
However,oneeasilyverifiesthatforallpointspwehavethatp⊥θ(p).Thusθ
doesnotmapanychambertoanoppositechamber,i.e.,itisnotaPhaninvolution.
Still,theoppositelinesl1=(12)(34)(56)andl2=(13)(26)(45)areinterchanged.
Hencethisisanexampleofanimproperflip.
Example2.1.11.Assumewearegivenaquasi-flipθofatwinbuildingC,anda
secondtwinbuildingCofsphericaltype(W,S).Thenθ×idisaquasi-flipofthe
buildingC×C,withminimalθ-codistanceequaltotheminimalθ-codistanceofθ
timesthelongestelementinW.Thisshowsthatfortwinbuildingswhicharenot
irreduciblethereisnoboundonthesizeoftheminimalθ-codistance.

-quasi-flipsNBSofaroursetupwasapurelygeometricone.Wedescribed(quasi-)flipsasbeing,
uptoatypechange,isometriesoftheinvolvedtwinbuildings.However,ourmain
motivationtostudyflipsistheirapplicationtogroups,i.e.,toautomorphismgroups
buildings.wintofphismsTherefore,ofgroupswenowwithinatrotwinduceBNthe-pair.conceptInofSectionBN2.2w-quasi-flips,ewillasademonstrateclassoftheautomor-close
correspondencebetweenBN-(quasi-)flipsandbuilding(quasi-)flips,justifyingthe
names.ofhoicecsimilarDefinition2.1.12.LetGbeagroupwithatwinBN-pair(B+,B−,N).Anauto-
morphismθofGiscalledaBN-quasi-flipif
(1)θ2=idand

22

2.1.BuildingflipsandBN-flips

(2)thereexistsg∈Gsuchthatθ(B+)=gB−g−1.
Remark2.1.13.LetFbeafield.LetGbeaconnectedreductivealgebraicF-group.
Ifθisaninvolutory(abstract)automorphismofG(F)(thegroupofF-rationalpoints
ofG),theconditionthatθ(B+)beconjugatetoB−isinfactalwayssatisfied,cf.
6.1.3.actFForsplitKac-Moodygroups,theconditionreducestoadichotomy:Eitherθ(B+)
isconjugatetoB−oritisconjugatetoB+,cf.Fact6.2.1inChapter6.
Example2.1.14.LetFbeafieldandG=SLn(F)thespeciallineargroupover
thisfield,consideredasamatrixgroup,andendowedwiththetwinBN-pairB+and
B−ofupperandlowertriangularmatrices.ThentheChevalleyinvolution,which
sendseveryelementx∈Gtoitstransposedinversetx−1,isaclearlyaninvolution.
ItalsointerchangesB+andB−,thelatterbeingconjugatetotheformer.Hencethis
exampleconstitutesaBN-quasi-flipasdefinedabove.
WenowshowthattheseeminglyweakconditionsofDefinition2.1.12implyfora
largeclassofgroupsthataBN-quasi-flipnotonlymapsB+toaconjugateofB−and
viceversabutevenmapsthemtosimultaneousconjugates.Asaconsequenceitalso
inducesanautomorphismoftheCoxetersystem(W,S).ThisiscentralinSection2.2
toprovethateveryBN-quasi-flipinducesabuildingquasi-fliponthetwinbuilding
associatedtothetwinBN-pair.Tosimplifytheexposition,werestrictourselvesto
saturatedtwinBN-pairs,butinviewof[AB08,Remark6.83andfollowing],this
restrictioniseasilyovercome.
Proposition2.1.15.LetGbeagroupwithsaturatedtwinBN-pair(B+,B−,N)of
type(W,S),letT:=B+∩B−.Letθbeaquasi-flipofG.Ifthesetofchambers
fixedbyTofthetwinbuildingassociatedtoGequalsthetwinapartmentcontaining
B+andB−,thenthefollowinghold:
(1)Thereexistsx∈Gsuchthatθ(Bε)=xB−εx−1andθ(x)x∈T,whereε∈
.,+−}{(2)θinducesauniqueautomorphismoftheCoxetersystem(W,S)oforderatmost
2(soitnormalizes−the1setS).Specifically,W=∼NT/Tandtheautomorphism
isgivenbynT→xθ(n)xT.
Proof.ThenormalizerNG(T)actsonthesetofchambersfixedbyTofthetwin
buildingassociatedtoG.Sincebyhypothesisthissetequalsthetwinapartment
containingB+andB−andsinceNequalsthefullstabilizerofthistwinapartment
(as(B+,B−,N)issaturated,cf.[AB08,Definition6.84]),theequalityN=NG(T)
holds.

2(1)Rethatcallθ(Bthat+)b=ygB−gdefinition−1.ofMoreoaverquasi-flip,bytheθ=Birkhoidffanddecompthereosition,existsg∈thereGsucexisth

23

Flips2.

b+∈B+,b−∈B−andn∈Nsuchthatθ(g)g=b+nb−.Then
θ(gTg−1)=θ(g(B+∩B−)g−1)
=θ(g)θ(B+)θ(g−1)∩θ(gB−g−1)
=θ(g)gB−g−1θ(g)−1∩B+
=(b+nb−)B−(b+nb−)−1∩B+.
Henceforx:=b+−1θ(g)wehave
xθ(T)x−1=b+−1θ(gTg−1)b+=nB−n−1∩B+≥T,
wherethelastcontainmentholdsbecauseofn∈N.Therefore
T≤xθ(T)x−1≤xθ(xθ(T)x−1)x−1=xθ(x)T(xθ(x))−1.
Accordingly,asTfixesauniquetwinapartment,T=xθ(x)T(xθ(x))−1,i.e.,
xθ(x)∈NG(T).Asanimmediateconsequenceθ(T)=xθ(T)x−1.Wenote
B+=xθ(g)−1b+B+b+−1θ(g)x−1=xθ(B−)x−1thus,
B+∩B−=T=xθ(T)x−1=xθ(B+)x−1∩xθ(B−)x−1=xθ(x)B−(xθ(x))−1∩B+.
SinceB−istheuniquechamberoppositeB+inthetwinapartmentfixedby
T,thismeansxθ(x)∈NG(B−)=B−and,inparticular,θ(x)B−=x−1B−.
Thereforexθ(x)∈B−∩NG(T)=Tandθ(B+)=θ(x)B−θ(x)−1=x−1B−x.
(2)LetX:={x∈G|θ(B+)=xB−x−1andθ(B−)=xB+x−1}.By(1),this
setisnonempty.Forx∈X,defineθx:g→x−1θ(g)x.Clearlyθxpreserves
T=B+∩B−,henceN,thusitinducesanautomorphismonW=N/Tby
sendingnTtoθx(nT).
ThisautomorphismonWdoesnotdependonthechoiceofx:Forifx∈X,
thenθx(g)=xx−1θx(g)xx−1.Butxx−1∈NG(B+)∩NG(B−)=B+∩B−=T.
ThusforallnT∈N/Twehaveθx(nT)=θx(nT).
ItremainstobeshownthatθxnormalizesS.Foreachs∈SthesetPs:=
B+∪B+sB+isarank1parabolicsubgroupofpositivesignofG.Letnsbea
representativeofsinN.Then
θx(Ps)=θx(B+)∪θx(B+)θx(ns)θx(B+)=B−∪B−θx(ns)B−
isaparabolicsubgroupofnegativesignofG:Itisagroupbecauseitisthe
imageofasubgroupofGunderthegroupautomorphismθx;itisparabolic
becauseitcontainstheBorelgroupB−.Sinceitconsistsofpreciselytwo
Bruhatdoublecosets,itmustagainbearank1parabolicsubgroup.Hence
θx(ns)isarepresentativeofsomes∈S.Assandsareindependentofthe
choiceofns,themapθxpermutesS.

24

2.1.BuildingflipsandBN-flips

Remark2.1.16.IfthegroupinProposition2.1.15isendowedwithalocallysplit
RGD-systemoverfields(Kα)α∈Φsatisfying|Kα|≥4foreachα∈Φ,thenby[Cap09,
Lemma4.8]thesetofchambersfixedbythetorusTequalsthetwinapartment
containingB+andB−.
Onthegrouptheoreticlevel,thisconditiononTisequivalenttoaskingthat
wheneverTg≤Tforsomeg∈Gthenwealreadyhaveg∈N.Thisisforexample
thecasewhenN=NG(T)andTisfinite.

Analogtobuilding(quasi-)flips,aBN-quasi-flipisaBN-flipifitistypepreserv-
ing:quasi-flipDefinitionθofG2.1.17.isaBLetNG-flipbifeathegroupinducedwithaautomotwinBrphismNN-pairT/(TB:+,nBT−,→Nx).−1θA(nB)xNT-
istrivial,whereT:=B+∩B−andx∈Gsuchthatθ(Bε)=xB−εx−1forε∈
.,+−}{Example2.1.18.TheChevalleyinvolutiondescribedinExample2.1.14isactually
NanareexampletheofamonomialBN-flipasmatrices,itcenandtralizesTthethegroupdiagonalW:FmatricesorweinhavGe.WSo=wNe/T,computewhere
whenθnormalizestheCoxetergroup:
θ(nT)=nT⇐⇒tn−1T=nT⇐⇒tnn∈T.
Butforamonomialmatrixn,onereadilyverifiesthattnn∈T.
W,Forassumeannexample>2isthatevenisandnottletypJe∈Gpreserving,denotei.e.,thedomatrixesnotwithcentronesalizeonthetheangroupti-
diagonalandzeroselsewhere.AsJhasorder2,conjugationbyJisaninvolutory
toaautomorphismconjugate.ofG.FinallyBeing,thisaninnerautomorphismautomorphism,doesnotitiscenobvioustralizethatW.B+isConsidermappfored
examplethecasen=4,then
11−1111−111
111111=1−1.
Againinanalogytobuildingquasi-flips,wedefinethenotionofaproperquasi-flip.
θ(hB+Definitionh−1)=hB2.1.19.−h−1W.ecallaBN-quasi-flipproperifthereexistsh∈Gsuchthat
chambers.Geometrically,theabovemeansthatθinterchangesthestabilizersoftwoopposite
andExampleB−,hence2.1.20.ispropTheer.ChevButalleyasininvtheolutioncasefromofbuExampleilding2.1.14quasi-flips,intercnothangesallBBN+-
quasi-flipsareproper.Forexample,thesymplecticbuildingflipfromExample2.1.9
next(2.1.9)issection),impropwhicer,handthencanbenecessarilyliftedistoaimpropBNer.-flipofSLn(F)(asisdetailedinthe

25

Flips2.

2.2.CorrespondencebetweenbuildingandBN-flips
Inthis(quasi-)flips,sectionweestablishinginvestigatetheclosetherelationconnectionbetwbeteenweenbuildingthetwo(quasi-)flipsconceptsandwhichBNw-e
laterfrequentlyexploitinordertoapplytoolsfromgeometrytosolvegrouptheoretic
ersa.vviceandproblemsThebulkofthissectionisjointworkwithRalfGramlichandBernhardMühlherr.

WestartbyshowingthateveryBN-flipinducesabuildingflipinanaturalfashion.
Forthis,∼weuse−1thenaturalisomorphismC+=∼G/B+,andimplicitlyalsousethat
G/B+={gB+g|g∈G}(seee.g.[AB08,Section6.2.4]).
aProptwinBositionN-pair2.2.1with(Seeassoalsociatedtwin[DMGH09,buildingPropC.ositionThen3.4])any.LBetNGbe-quasi-flipagroupθofwithG
inducesabuildingquasi-flipθ˜ofCbysendinggBεtoθ(g)xB−εforx∈Gand
ε∈{+,−},asinProposition2.1.15.Thisquasi-flipisuniquewiththeproperties
that(1)foranyg∈G,andanychamberc,wehaveθ˜(gc)=θ(g)θ˜(c);
(2)θ˜mapsthechamberstabilizedbyB+tothechamberstabilizedbyθ(B+).
(i.e.,Bothθtypeandprθ˜eserving)induceiftheandsameonlyifθ˜is.automorphismFurthermorof(We,,Sθ).isInpropperifarticular,andθonlyisifaθ˜flipis.
Proof.RecallfromRemark1.8.4thatCconsistsofthebuildingsG/Bεforε∈{+,−}
withdistance−1functionsδε:G/Bε×G/Bε→Wsatisfyingδε(gBε,hBε)=wif
δεand:(onlyG/B+if×BεgG/B−hB)ε∪(=G/BBεw−B×ε.G/BThese+)→areWtwinnedsatisfyingbyδ∗(thegBεco,hB−distanceε)=wiffunctionand
onlyifBεg−1hB−ε=BεwB−ε.
1−{+,By−},Propandθosition(x)x∈2.1.15,T.thereDefineaexistsxbijectiv∈eGmapsuchθ˜bthatetwθ(eenBε)=G/B+xB−εandxG/Bfor−εb∈y
sendinggBεtoθ(g)xB−ε.Thismapiswell-definedandhasorder2since
θ˜(θ˜(gBε))=θ˜(θ(g)xB−ε)=θ(θ(g)x)xBε=gθ(x)xBε=gBε.
Defineautomorphismθx:g→ofx−the1θ(Wg)eylx.groupAgainWby=NProp/TositionvianT2.1.15→θ(thisnT)=inducesθ(n)aT.weNowll-defined
xxδε(gBε,hBε)=w⇐⇒Bεg−1hBε=BεwBε
⇐⇒θ(Bε)θ(g−1h)θ(Bε)=θ(Bε)θ(w)θ(Bε)
⇐⇒B−εx−1θ(g−1)θ(h)xB−ε=B−εx−1θ(w)xB−ε=B−εθx(w)B−ε
⇐⇒δ−ε(θ(g)xB−ε,θ(h)xB−ε)=θx(w).
thatfindewSimilarlyδ∗(gBε,hB−ε)=w⇐⇒δ∗(θ(g)xB−ε,θ(h)xBε)=θx(w).

26

2.2.CorrespondencebetweenbuildingandBN-flips

Therefore,ourBN-(quasi-)flipinducesabuilding(quasi-)flipwiththeclaimedprop-
erties.Uniquenessfollowsreadily.Itisalsoclearthatbothinducethesameauto-
morphismof(W,S).
Finally,ifθisproper,thenthereisb∈GsuchthatbB+b−1ismappedtobB−b−1.
ThenthechambersbB+andbB−areopposite,andareinterchangedbyθ.
Wenowturntotheconversequestion:GivenagroupGwithtwinBN-pairanda
building(quasi-)flipθontheassociatedbuilding,isthereaBN-(quasi-)flipinducing
θ?IfGisgeneratedbyitsrootgroupsandactsfaithfullyonthebuilding,theanswer
isyes,asthefollowingtheoremmakesprecise.
Theorem2.2.2(jointworkwithGramlichandMühlherr).Let(G,{Uα}α∈Φ,T)
beareducedRGD-systemoftype(W,S)suchthattheassociatedtwinbuildingis
strictlyMoufang(e.g.itsdiagramcontainsnoisolatednodes).Thenany(proper)
buildingquasi-flipofCinducesa(proper)BN-quasi-fliponG.Bothinducethesame
automorphismof(W,S).
Proof.Supposeθisaquasi-flipofC.Thenitinducesanautomorphismθ˜ofAut(C)
byconjugation:Ifg∈Aut(C)isanautomorphismofthebuilding,thenθ◦g◦θis
againabuildingautomorphism.WeassumedtheRGD-systemtobereduced,soG
actsfaithfullyonCandisgeneratedbyitsrootgroups.SinceweassumedCtobe
strictlyMoufang,by[AB08,Theorem8.81andProposition8.82],Giscanonically
isomorphictothesubgroupG†ofAut(C)generatedbytherootgroupsofAut(C).
Sinceθnormalizesthesetoftwinapartmentsresp.thesetoftwinroots,wededuce
thatθ˜normalizesG†.Henceθ˜∈Aut(G).AllpropertiesofaBN-quasi-flipfollow
readily.Itisclearthatθ˜istypepreservingifθis.
Ifθisproper,thereexistsachambercwhichismappedbyθtoanopposite
chamberd.SincethediagonalactionofGbyleftmultiplicationonG/B+×G/B−
istransitiveonthepairs˜ofoppositechamb−1ers,wecan−find1h∈Gsuchthatc=hB+
andd=hB−.Thus,θinterchangeshB+handhB−h.
Anaturalquestionnowiswhatcanbesaidaboutliftsofquasi-flipstoanon-
reducedRGD-systems(G,{Uα}α∈Φ,T).Here,thingsarenotquiteasnice.Inpar-
ticular,fornon-centeredRGD-systems,many“wild”thingscanhappen,e.g.com-
plicatedgroupextensionsmaybeinvolvedforwhichtheexistenceofaliftingofa
givenflipisfarfromclear.
Butevenwhenrestrictingtocenteredbutnon-faithfulRGD-systems,wewould
havetobeabletoliftθtoarbitrarycentralextensionsofG.Specifically,Theorem
2.2.2impliesthatgivenanRGD-systemandabuildingquasi-fliponitsassociated
twinbuilding,wecanalwaysliftthebuildingquasi-fliptoaBN-quasi-fliponthe
reductionoftheRGD-system.ButsupposeGisacentralextensionofthegroupof
thereducedRGD-system.Thentogetaquasi-fliponGwehavetoknowhowto
liftthebuildingfliptothecenterofG,whichmaynotbepossibleingeneral.In
summary,nogeneralanswertothisproblemisknowntous,andwebelieveittobe
averydifficultproblemingeneral.

27

Flips2.

However,itisatleastpossibletoliftquasi-flipstouniversalcentralextensionsof
GD-systems:RteredcenCorollary2.2.3.GivenacenteredRGD-system(G,{Uα}α∈Φ,T),letCbetheas-
sociatedtwinbuilding.AssumethatGisperfect.Thenanybuildingquasi-flipofC
inducesaBN-quasi-flipontheuniversalcentralextensionofG.
Proof.Assumewearegivenabuildingquasi-flipθofC.Sinceweareinthecentered
case,by[AB08,Propositions8.82(2)],thekerneloftheactionofGonCcoincides
withitscenter.HencewegetareducedRGD-systemforG/Z(G),andbyProposition
2.2.1wecanliftthebuildingquasi-flipθtoaBN-quasi-flipθofG/Z(G).
extensionG(see[Mil72,Chapter5]formoreonuniversalcentralextensions).Let
SinceGisperfect,alsoG/Z(G)isperfect.ThusG/Z(G)admitsauniversalcentral
π:G→G/Z(G)betheassociatedcoveringmap.Thenθ◦πalsoisacoveringmap
ofG/Z(G),hencebytheuniversalityproperty,thereexistsanautomorphismθofG
suchthatθ◦π=π◦θ.Sinceθisaninvolution,weevenhave
π=θ2◦π=θ◦π◦θ=π◦θ2.
Thisimpliesthatθ˜isaninvolutionaswell.
Finally,weobtainatwin-BN-pairofGbytakingthepreimagesunderπofB+,
B−,T,Nandsoon.(Thatthisisagainatwin-BN-pairisreadilyverified,aswe
onlyreplacedeverythingbycentralextensions.)Clearlyθ˜isaquasi-flipwithrespect
tothistwin-BN-pair.

Giventhecorrespondenceestablishedinthissection,thesimilarchoiceofnames
forbuilding-andBN-quasi-flipsisfinallyjustified.
Also,wheneverwestartwithaBN-(quasi-)flips,thereisauniquecorresponding
building(quasi-)flip,sointhiscasethereisnoneedatalltodistinguishbetween
thetwonotions.Thisalsomeansthatanyconceptwedefineforbuildingquasi-
flipscanbeimmediatelytransferredtoBN-quasi-flips.Namely,wemaysaythat
aBN-quasi-fliphaspropertyXexactlywhentheassociatedbuildingquasi-fliphas
X.yertpropWewillmakefrequentandliberaluseofthisfactinsubsequentsections,usually
bynotdistinguishingbetweenbuilding-andBN-quasi-flipsexplicitlyandinstead
simplyusingtheterm“quasi-flips”.

tdescenepSte2.3.Inpreparationforfurtherworklateron,wenowtakeacloserlookatthepossibil-
itiesfortheθ-codistancesthatcanoccur.Fromthiswederiveagenericreduction
argumentthatallowsustotransfermanyquestionsfromthetwin-buildingcontext
tothesimplerandmorerestrictedcontextofsphericalbuildings.

28

tdesceneepSt2.3.

Lemma2.3.1.Letcbeanarbitrarychamber,letw:=δθ(c)beitsθ-codistance.
Thenwisaθ-twistedinvolution,thatis,θ(w)=w−1.Inparticular,ifθisaflip,
thenw=θ(w)andwisaninvolution.
Proof.By(Tw1)wehaveδ∗(θ(c),c)=w−1.ApplyingLemma2.1.4,weobtain
w=δ∗(c,θ(c))=θ(δ∗(θ(c),c))=θ(w−1).
Wenextgiveaconditionunderwhichachamberadmitsaneighboringchamber
withlowernumericalθ-codistance,i.e.,achamberwhichisfartherawayfromits
imagethantheoriginalchamber.
Lemma2.3.2.Letcbeanarbitrarychamberwithθ-codistancew:=δθ(c).Assume
s∈Ssatisfiesl(swθ(s))=l(w)−2.Thentheθ-codistanceofallchambersin
Ps(c)\{c}equalsswθ(s).
Proof.Letd∈Ps(c)\{c}.Sincel(sw)=l(wθ(s))=l(w)−1,thesecondtwin
buildingaxiom(Tw2)impliesδ∗(c,θ(d))=wθ(s).Anotherapplicationof(Tw2)
yieldsδ∗(d,θ(d))=swθ(s).
Thefollowingisanextensionof[GM08,Lemma2](there,however,noproofis
en).givLemma2.3.3.Letr∈Wbeaθ-twistedinvolution,andw∈Wsuchthatl(w−1rθ(w))=
l(r)−2l(w).
(1)Letc∈Cwithδθ(c)=r,andletdbeachamberatdistancewfromc.Then
δθ(d)=w−1rθ(w).
(2)Letd∈Csuchthatδθ(d)=w−1rθ(w).Thenthereexistsauniquechamberc
withdistancew−1fromdsuchthatδθ(c)=r.
(3)Ineithercase,theconvexhullofdandθ(d)containscandθ(c).
Proof.Wefixaminimaldecompositions1∙∙∙snofw.
(1)Pickaminimalgallery(c=c0∼s1c1∼s2∙∙∙∼sncn=d)joiningcandd.
Thusbyhypθothesis,l(s1r)=l(r)−1=l(rθ(s1)).ApplyingLemma2.3.2,we
concludethatδ(c1)=s1rθ(s1).Repeatingthisargument,weget
δθ(d)=(sn∙∙∙s1)∙r∙θ(s1∙∙∙sn)=w−1rθ(w).
(2)Theclaimfollowsbyinductiononn.Forn=0thereisnothingtoshow.So
supposen>0.ByAxiom(Tw3)thereisauniquechamberdwhichissn-
adjacenttodsuchthatδ∗(d,θ(d))=snw−1rθ(w)=−(1sn−1∙∙∙s1)∙−r1∙θ(s1∙∙∙sn).
Thiscannotbea−1θ-twistedinv−1olution(aselse,snwrθ(w)=wrθ(wsn),im-
plyingthatsnwrθ(wsn)=wrθ(w),contradictingthehypothesis).There-
foreδθ(d)=snw−1rθ(w)θ(sn)=(sn−1∙∙∙s1)∙r∙θ(s1∙∙∙sn−1).

29

Flips2.

(3)Thiscanbeseenasaconsequenceof(2).However,wegiveanalternative
proof,oncemorebyinductiononn.Forn=0,nothinghastobeshownas
c=d.Supposenown>0,andtakethesamegallerybetweencanddasin
(2).LetPbethesn-panelaroundd.Wehaveδ∗(d,θ(d))=w−1rθ(w)but
δ∗(cn−1)=snw−1rθ(w).Thus,projP(θ(d))=cn−1.Itfollowsthattheconvex
hullofdandθ(d)containscn−1andbysymmetryalsoθ(cn−1),hencealsotheir
convexhull.Bytheinductionhypothesis,wearedone.
ByProposition1.3.3,θ-twistedinvolutionsareconjugatetothelongestelement
ofsomesphericalstandardparabolicsubgroupsoftheWeylgroupW.Wecombine
thiswithLemma2.3.3towalk(or,asIliketoputit,“descend”)fromarbitrary
chamberstochamberswithsphericalθ-codistance.
Lemma2.3.4.Letc∈C+beanarbitrarychamberwithθ-codistancew.Thenthere
existasphericalsubsetIofSandanelementw∈Wsuchthatthefollowinghold:
(1)w=wwIθ(w)−1andl(w)=2l(w)+l(wI),wherewIisthelongestelementof
.WI(2)w≤wandwI≤wintheBruhatorder.
(3)Everychamberd∈C+withδ+(c,d)=wsatisfiesδθ(d)=wI.
(4)Theconvexhullofdandθ(d)containscandθ(c).

flipsStrong2.4.Inthissectionwepresentaparticularlywell-behavedclassofquasi-flips,thestrong
flips.Manyresultswhichareprovenwithmuchlaborinthisthesisbecomemuch
simplerwhenonerestrictstothisclass;conversely,featuresoftheseflipsmayinspire
neralizations.geossiblepThefollowingdefinitionisessentiallytakenfrom[DM07,Definition6.2];butnote
thatinloc.cit.,whatwecallflipiscalledinvolutionandwhatwecallstrongflipis
flip.calledjustDefinition2.4.1.Letθbeabuildingquasi-flipofatwinbuildingC.Forany
sphericalresidueR,definetheset
projR(θ):={c∈R|projR(θ(c))=c},
whereprojRdenotestheprojectionontoR.IfforallpanelsPofCwehaveprojP(θ)=
P,wecallθastrongquasi-flip,andsaythatithastheDevillers-Mühlherr
.yertpropWeremindthereaderthatinviewofProposition2.2.1,wecannowalsotalkabout
-quasi-flips.NBstrong

30

2.5.Stabletwinapartments

Theimportanceofthisdefinitionistwo-fold:Firstly,manyargumentscanbe
considerablysimplifiedforstrongflips,e.g.strongerdescentpropertieshold,asthe
followinglemmaillustrates,orasalookatthebeautifulfiltrationresultof[DM07]
willreveal.Secondly,manyinterestingquasi-flipsareactuallystrong,makingit
worthwhiletostudythemspecifically.
Lemma2.4.2.Letθbeastrongquasi-flip,letcbeachamberwithθ-codistancew.If
s∈Sissuchthatl(sw)<l(w),thenthereexistsachamberdwhichiss-adjacentto
candhaslowernumericalθ-codistance.Inparticular,strongquasi-flipsareproper.
Proof.Ifw=1Wnothinghastobeshown.Otherwise,takeanys∈Ssuchthat
l(sw)<l(w)andconsiderthes-panelPcontainingc.ByLemma1.3.2,forevery
chamberinP,theθ-codistancecanonlybew,sw(whichthenequalswθ(s))or
swθ(s),allofwhicharelessorequalwintheBruhatorder.ThusprojP(θ(c))=
c.ButbytheDevillers-Mühlherrproperty,thereexistsachamberdinPsothat
projP(θ(d))=d.Thusthenumericalθ-codistanceofdisstrictlylowerthanthatof
c.Inparticular,wecanrepeatthisprocessuntilwereachaPhanchamber.
Example2.4.3.Theprototypicalexampleofastrongflipisthefollowing:Suppose
Fisafieldendowedwithanontrivialfieldinvolutionσ(e.g.thecomplexnumbers
withcomplexconjugation,orafinitefieldofsquareorderwiththecorresponding
poweroftheFrobeniusautomorphism).SeealsoLemma6.1.12.
Throughoutthepresentwork,wewilloccasionallymentionwhenresultsholdfor
strongflips,oraresimplertoproveforthem.

2.5.Stabletwinapartments
Inthissectionweproveundersomemildconditionstheexistenceofθ-stable(twin)
apartmentsaroundanychamberc.Hereasusualθisaquasi-flipofaMoufangtwin
buildingC.Fromthiswederiveanicedoublecosetdecompositionofgroupswitha
twinBN-pairadmittingaBN-quasi-flip.
Theconditionwearegoingtoimposewillbethatallrootgroupsareuniquely
2-divisible.Thisgeneralizestheideaofagroupbeingdefinedoverafieldofchar-
acteristicdifferentfrom2,inthesensethatallalgebraicandKac-Moodygroups
definedoversuchafieldsatisfyit.
Definition2.5.1.Letnbeanintegergreaterthan1.AgroupGiscalledn-
divisibleifforeachg∈Gthereexistsh∈Gsuchthathn=g.Ifhisuniquewith
thatproperty,wecallGuniquelyn-divisible.
NotethatwedonotrequireGtobeabelian,asisusuallythecaseintheliterature
whendefiningn-divisibility.Also,oftenintheliterature,nisrequiredtobeprime.
ButclearlyGisn-divisibleifandonlyifGisp-divisibleforeachprimepdividingn.
Thefollowingpropositionisoneofthekeyingredientsofthemainresultofthis
section.

31

Flips2.

Proposition2.5.2.LetM=(X,(Ux)x∈X)beaMoufangset.Iftherootgroups
Uxareuniquely2-divisible,thenaninvolutoryautomorphismofMfixingapoint
necessarilyfixesasecondpoint.
Proof.Supposeφisaninvolutoryautomorphism(i.e.,apermutation)ofMwhich
fixesapoint,say∞;soU∞φ=U∞.AsinSection1.10,wewillalwaysdenotethe
actionofapermutationonasetontheright,i.e.,wewillwritea.φratherthanφ(a).
LetabeanyelementofMdifferentfrom∞.Ifa=a.φ,wehavefoundasecond
fixedpointandaredone.Soassumea=a.φ.SinceU∞actssimplytransitivelyon
X\2{∞},thereexistsauniqueg∈U∞suchthata.g=a.φ.Chooseh∈U∞such
thath=g.Weclaimthata.hisafixedpoint.Indeed
(a.g).g−1=a=(a.g).φ=a.gφgφ=(a.g).gφ.
SinceU∞actssimplytransitively,wehavegφ=g−1,andasU∞isuniquely2-divisible
thisimplieshφ=h−1aswell.Therefore
(a.h).φ=(a.φ).hφ=(a.g).hφ=(a.h2).h−1=a.h.
Remark2.5.3.Forabelianrootgroupsthestatementabovecanbeeasilyextended
tofiniteautomorphismgroupsΓ:IfΓfixesonepointandtherootgroupsare|Γ|-
divisible,thenΓfixesasecondpoint.Itisaninterestingquestionwhetheronecan
extendthistonon-abelianrootgroups.
AnalternativewayofstatingProposition2.5.2isthataninvolutoryautomorphism
ofarank1buildingwith2-divisiblerootgroupswhichfixesachamber,alsofixes
anoppositechamber.ThefollowingpropositionextendsthistosphericalMoufang
buildingsofhigherrank,fromwhichtheexistenceofθ-stableapartmentsfollows
.immediatelyProposition2.5.4.GivenasphericalRGD-system(G,{Uα}α∈Φ,T),letCbethe
associatedsphericalbuilding.AssumeallrootgroupsUαareuniquely2-divisible.Let
θbeaquasi-flipofGwhichfixessomeBorelsubgroupB.ThenthereexistsaBorel
subgroupBoppositeB(i.e.,BandBintersectinatorus)whichisfixedbyθ.
Geometrically,letcbethechambercorrespondingtoB,thenthereexistsachamber
cfixedbyθandoppositec.
Proof.Inthefollowing,wetakethegeometricviewpoint,whereitiseasiertoargue.
So,θisabuildingquasi-flipofCinthesenseofRemark2.1.2(resp.analmost
isometry,asdefinedin[AB08,Section5.5.1]).Thenθinducesanautomorphismof
(W,S)oforderatmost2.DenotebyIthesetofθ-orbitsinS.ForeachI∈I
weshowthattheresidueRI(c)containsachambercIfixedbyθandoppositecin
thatresidue,i.e.,δ(c,cI)=wI,wherewIdenotesthelongestelementofWI=I.
If|I|=1,thisisProposition2.5.2.Soassume|I|=2,say,I={s,t}.ThenRI(c)
isaMoufangn-gonwhichisnormalizedbyθ.Weconstructagallery(c0,...,cm−1)
oflengthm:=n2+1withc0:=c:Forc1chooseanychamberdifferentfrombut

32

2.5.Stabletwinapartments

sbutt-adjacen-adjacenttotc0to.cIf1,mand=2so,stopon,ahere;lternatingelse,cbhoetwoseeenfors-c2andanytcham-adjacenbertchamdifferenbters.from
iseven,Considerthenno2wm−the1θ=n-stable+1,andgalleryc(θ(cism−1opp),..osite.,c,θ(.c..,c)m,−1)andofforlengththat2mreason−1.Iftheyn
m−1m−1
spanaθ-stableapartmentcontainingc.(SeeFigure2.1a.)
Ifnisodd,somemoreeffortisneeded.LetP:=Pt(cm−1)bethet-panelcontaining
cm−1.Thenbyconstruction,Pandθ(P)areoppositepanelsinRI(c).Bycomposing
θandtheprojectionmapfromθ(P)toP,weobtainanautomorphismθ=projP◦θ
oftheMoufangsetPoforderatmost2.Clearlyθfixescm−1.HencebyProposition
2.5.2thereisasecondchambercm∈Pfixedbyθ.Butthen(θ(cm),...,c,...,cm)
isaθ-stablegalleryoflength2m+1=n+2,andcmisoppositetoθ(cm−1),andthe
twoInofeitherthemcase,spanweaθobtain-stableaθ-stableapartmentconapartmentainingtconc.taining(Seec,Figurewhich2.1b.)thennecessarily
conNowtainstheauniquelongestchamelemenbetrw0oppofositeWccanandbealsowrittenfixedasbayθpro.ductofthelongestwords
wabIo,vIe∈wIecan(seefind[Ste68a,acham1.32]).berdSofixedassumebywθ0at=wI1distancewI2∙∙∙wwIk.fromStarc.tingWeinproc,bceedytheto
I1findachamberd2fixedbyθandatdistancewI2fromd1,1hencedistancewI1wI2from
c.Werepeatthisuntilwefinallyreachachamberdk,fixedbyθandatdistance
w0fromc,i.e.,oppositec.Thisyieldsthedesiredθ-stableapartmentsinceopposite
chambersdetermineauniqueapartment.
Example2.5.5.Toillustratethe2-divisibilitycondition,wesketchanexample:Let
GbeasplitalgebraicgroupoverafieldF,andCtheassociatedsphericalMoufang
building.ThenallrootgroupsareparametrizedbytheadditivegroupofF.Hence
theyareuniquely2-divisibleifandonlyifcharF=2.
InarbitrarysphericalMoufangbuildingstheclassificationofMoufangpolygons
shofields;wsthathencewhereegetallarootsimilargroupsareconditioninaonsensethecadditivharacteristicegroupsofofsome(vectorunderlyingspacesovfield.er)
WemakethispreciseinSection2.6.
Remark2.5.6.TheconditionontherootgroupsinPropositions2.5.2and2.5.4is

(a)Moufangquadrangle(b)Moufangprojectiveplane
Figure2.1.:Constructingaθ-stableapartmentinsideMoufangpolygons.

33

Flips2.

essentialinthefollowingsense:TakeanysphericalMoufangbuildingCassociated
tosomeFq-locallysplitRGD-systemwithq=2n.Therootgroupsthenarenot2-
divisible,theyevenadmit2-torsion.Letαbeapositiveroot,andtakeanarbitrary
nontrivialelementuinUα.ThenuisaninvolutoryautomorphismofC,fixingthe
chambercstabilizedbyB+.NowuactsonthesetP\{c},wherePisanypanel
intersectingtherootαonlyinc.ButPhasoddsizeq+1,andweknowthatu
fixesc.AsUαactssharplytransitively,ucannotfixanyotherchambersinP.In
particular,ucannotfixanyapartmentinC.
Ontheotherhand,the2-divisibilityconditionisnotstrictlynecessary:Takethe
Fanoplane,theprojectiveplaneoverF2.Thisprojectiveplaneadmitsanupto
isomorphismuniquepolarity,whichthenisabuildingflip.Infactitisaproper
buildingflipandonereadilyverifiesthateachchamberiscontainedinanapartment
stabilizedbythepolarity.Infactthisgeneralizestoarbitrarypolaritiesofprojective
planesincharacteristic2,usingargumentssimilartothoseusedinSection4.6.3.
Thekeyobservationhereisthatinthissituation,everylinecontainsatleastone
absolutepoint(see[Bae46,Theorem1]forthefinitecase,whichcanbegeneralized
tothegeneralcaseusingMoufangsetarguments).
Remark2.5.7.ThestatementsofPropositions2.5.2and2.5.4wereinspiredby
[Müh94,Section3.5](sadly,thisthesiswasneverpublishedandhenceisdifficultto
obtainandnotaswell-knownasitshouldbe).Inloc.cit.,fixedpointsofanarbitrary
finitegroupofautomorphismsareconsidered.Inthepresentwork,wefocusonthe
specialcaseofasingleinvolutoryautomorphism.Thisenablesustoemploydifferent
methodsfortheproofsandgetsomewhat“better”results,atthelossofagreatdeal
ofgenerality.Togetaflavoroftheimprovement,herearetwoexamples:
First,applyingLemma3.5.4fromloc.cit.totherootgroupsofasuitableMoufang
setyieldsaresultsimilarinspirittoProposition2.5.2.Butbyspecializingtothe
case|Γ|=2,weareabletoreducetheassumptionsonehastoimposeontheroot
groups;inparticular,nonilpotencyhastobeassumed.
Secondly,Theorem3.5.5ofloc.cit.isverysimilartoProposition2.5.4.However,
theconditionsimposedtherearelessexplicitandlesspracticalthanours.For
example,theunipotentradicals(cf.Definition1.9.5)oftheBorelsubgroupsmust
satisfycertainfiltrationconditions,whicharenotknowningeneral.Comparedto
this,2-divisibilityoftherootgroupsisinmanycasesknownoreasytoverify.
Wefinallyconcludeforanyflipθandanychamberctheexistenceofθ-stable
apartmentscontainingc,providedtherootgroupsareuniquely2-divisible.
Theorem2.5.8.Letθbeaquasi-flipofanRGD-system(G,{Uα}α∈Φ,T),letC
betheassociatedtwinbuilding.AssumeallrootgroupsUαareuniquely2-divisible.
ThenforanyBorelsubgroupBofG,thereexistsaθ-stableconjugateofTinB.
Geometrically,foranychamberc,thereexistsaθ-stabletwinapartmentcontaining
.cProof.ByLemma2.3.4thereexistasphericalsubsetIofSandachamberd∈C+
suchthatδθ(d)=wIandtheconvexhullofdandθ(d)containscandθ(c).

34

2.6.2-divisiblerootgroups

Sinceδθ(d)=wI,thesphericalI-residueRI(d)isoppositetoitsimageunderθ,
andthusisaPhanresidue.Therefore,ifwecomposeθwiththeprojectionmap
fromθ(RI(d))=RI(θ(d))toRI(d),weobtainaninvolutoryalmostisometryθof
thesphericalbuildingRI(d).Clearly,θfixesthechamberd.Wecannowapply
Proposition2.5.4tofindasecondchamberdinRI(d)fixedbyθandoppositedin
RI(d).Thatis,
δ+(d,d)=wI=δ∗(d,θ(d))=δ∗(d,θ(d)),
thereforeδ∗(d,θ(d))=1W.Itfollowsthattheconvexhullofdandθ(d)definesan
apartmentΣ.Onereadilyverifiesthatalsodandθ(d)arecontainedinΣ,which
henceisθ-stableandcontainsc.

2.6.2-divisiblerootgroups
InsplitthisRsection,GD-systems,weintvhisisestigateeasy:whenallroroototgroupsgroupsareareuniquelyisomorphic2tothe-divisible.additivForelogroupcally
ofnotthehaveundecrlyingharacteristicfield,2,andifandhenceonlyareifunithequelyroot2groups-divisibleareif2and-torsiononlyiffree.thefielddoes
pItolygons.turnsForoutthisthatweaexploitsimilarthestatementclassificationholdsofforroMoufangotgroupspoolygonsccurringgivenininMoufang[TW02].
Proposition2.6.1.LetM=(X,(Ux)x∈X)beaMoufangsetoccurringinaMoufang
polygon.IfU=U∞is2-torsionfree,thenitisuniquely2-divisible.
Pro[TW02,of.WeChapterfollow16].theWeexplicitrecommendentoumerationsimofultaneouslyallloMoufangokatploc.olygonscit.whilepresentedreadingin
.ofprothisItwillbecomeapparentthatallrootgroupsinMoufangpolygonsareessentially
ofeitheronetheofthese.additiveHencegroupofthearofield,otavgroupsectorwillspacebeov2era-torsionfield,orfreeasifub-andorsuponlyifergroupthe
underlyingfieldisnotofcharacteristic2.Wewillimplicitlyusethisfactbelow.
TrianglesT(A).Aisanalternativedivisionring,therootgroupsareparametrized
byfielditsK).additivHenceetheygroup,arewhichuniqueislyab2elian-divisible(iniffactanditisonlyavifcectorharAspace=coharverK=some2.
QuadranglesQI(K,K0,σ)ofinvolutorytype.Kisafieldorskew-field,σan
involutionofKandK0isanadditivesubgroupofKcontaining1.Twoof
therootgroupsareparametrizedbytheadditivegroupofK,theothertwoby
K0.IfcharK=2,thenxby[TW02,Remark11.2]wehaveK0=FixK(σ).So
x∈K0ifandonlyif2∈K0.Theclaimfollows.
QuadranglesspaceovQerQK(.K,TheL0,qro)otofgroupsquadraticareformparametrizedtype.Kbyistheafield,additivLe0isgroupavofectorK
resp.byL0.Againtheclaimfollowsreadily.

35

Flips2.

QuadranglesQD(K,K0,L0)ofindifferenttype.Kisafieldofcharacteristic2,
andK0andL0areadditivesubgroupsofKcontaining1,whichparametrize
therootgroups.Soallrootgroupsadmit2-torsion.
QuadranglesQP(K,K0,σ,L0,q)ofpseudo-quadraticformtype.Kisafieldor
askew-field,L0isarightvectorspaceoverK.Also,qisananisotropicpseudo-
quadraticformonL0(see(11.16)and(11.17)inloc.cit.)Following(11.24)in
loc.cit.,wedefinethegroup
T={(a,t)∈L0×K|q(a)−t∈K0}
withgroupoperationgivenby
(a,t)+(b,u):=(a+b,t+u+f(b,a)),
wherefisaskew-hermitianformonL0suchthat
q(a+b)≡q(a)+q(b)+f(a,b)(modK0).(2.1)
ThentherootgroupsareparametrizedbytheadditivegroupofKresp.byT.
NotethatThastheadditivegroupofKasasubgroup,i.e.,{0}×K≤T.So
weget2-torsionintherootgroupsifcharK=2.
SupposenowthatcharK=2.WeprovethatTisuniquely2-divisible:Given
anyelement(a,t)∈T,weeasilycomputetheuniqueelement(b,u)∈L0×K
suchthat(a,t)=2(b,u):
(b,u)=(a/2,t/2−f(b,b)/2)=(a/2,t/2−f(a/2,a/2)/2).
Toseethat(b,u)∈T,weusethatq(2b)=q(a)≡t(modK0)andcompute
q(b)−u≡q(b)−(t/2−f(b,b)/2)
(2.1)1≡2(2q(b)+f(b,b)−q(2b))≡0(modK0).
QuadranglesQE(K,L0,q)oftypeE6,E7andE8.Kisafield,L0isavector
spaceoverK.X0isanothervectorspaceoverK,andgsomefunctionfrom
X0×X0toK.LetSbethegroupwithunderlyingsetX0×Kandgroup
operationgivenby
(a,s)+(b,t)=(a+b,s+t+g(a,b))
foralla,b∈X0ands,t∈K.Therootgroupsarethenparametrized
bySandL0,whichare2-torsionfreeifcharK=2.Inthatcase,given
anarbitraryelement(a,s)∈S,aneasycomputationshowsthat(b,t):=
(a/2,s/2−g(a/2,a/2)/2)istheuniqueelementofSsatisfying2(b,t)=(a,s).

36

2.6.2-divisiblerootgroups

QuadranglesQF(K,L0,q)oftypeF4.Kisafieldofcharacteristic2andL0a
vectorspaceoverK.Furthermore,acertainsubfieldFofKisdefined(see
(14.3)inloc.cit.).ThentherootgroupsareparametrizedbyX0⊕Kand
W0irrelev⊕Fantfor(whereusXin0thisandWcon0aretext).certainSincevKectorandspaceshenceoFverareF,ofcwhichisharacteristichowever2,
allrootgroupsadmit2-torsion.
HexagonsspaceoHv(erJ,FF,,and#).theByroot[TW02,groupsareDefinitionparametrized15.16],Fbyisathese.fieldTheandclaimJafollovectorws.
(2)OctagonsK×KO(Kwhic,σh).hasKistheafieldadditivofecgroupharacteristicK=K2,×and{0}Kasσaisasubgroup.grouponThetherosetot
groupsareparametrizedbytheadditivegroupofKresp.byKσ(2),bothof
-torsion.2admithwhicItwouldbenicetohaveageneralargumentfortheabove,whichdoesnotrelyon
theclassificationofMoufangpolygons,andwhichmightbeapplicableinabroader
context.Inthefinitecase,thingsarequiteeasy.
Lemma2.6.2.LetUbeafinitegroup.ThenUisuniquely2-divisibleifandonly
ifUhasoddorder.
Proof.IfUhasevenorder,thenitcontainsaninvolutionx.Hencex2=1=12but
x=1,soUisnotuniquely2-divisible.
IfUhasoddorder,theneveryelementxhasoddorder,sayn=2k−1.Then
y:=xksatisfiesy2=x2k=x.Moreover,anyelementzwhichsquarestoxgenerates
acyclicgroupofoddorderwhichcontainsxandhencey.Butinsuchagroup,
squaringisagroupautomorphism,hencey=z.

Onemighthopetogeneralizethisideatoinfiniterootgroups.Anaturalidea
wouldbetogeneralize“oddorder”to“2-torsionfree”,inanalogytoProposition
whic2.6.1.hHothewroevoter,thegroupsfolloarewingabelianexampleand2shows-torsionthatfree,thereyetarenot2infinite-divisible.Moufangsetsfor
Example2.6.3.ConsideranyimperfectfieldFofcharacteristic2(e.g.thefield
F2((Moufangt))ofsetLaurenAGt(1,Fserie)shasintroootverthegroupsfiniteisomorphicfieldF2).toFThen∗,thewhichissharplyab2elian-transitivand2e-
torsionfreebutnot2-divisible:SquaringisjusttheFrobeniusmapofthisfield,
whichisnotsurjectiveinanimperfectfield(inourexample,thereisnosquareroot
).tofMorally,whatwelearnfromthissectionisthatfieldsofcharacteristic2cause
nothingbuttroublewhendealingwithflips.

37

Flips2.

2.7.Doublecosetdecomposition
Inthissectionwepresentadoublecosetdecompositionresult,generalizingresp.
adapting[HW93].There,F-involutionsofalgebraicgroupsareconsidered,whereF
isafieldofcharacteristicdifferentfrom2.Weextendthistoquasi-flipsofgroups
withatwinBN-pairwithuniquely2-divisiblerootgroups.Theresultsinloc.cit.
inturnrefineSpringer[Spr84],whichdealswithalgebraicallyclosedfields,andalso
Rossmann[Ros79]andMatsuki[Mat79]forF=R.Seealso[KW92]foraresulton
Kac-Moodygroupsoveralgebraicallyclosedfieldsofcharacteristic0.
Ourapproachisbasedprimarilyonbuildingtheoreticarguments,unlikeprevious
proofs.Thisallowsustotreatthesubjectinaunifiedway,andworksforarbitrary
quasi-flipswhichsatisfytheconclusionofTheorem2.5.8.Thisextendspreviouswork
invariousways.Forexample,inthecontextofalgebraicgroupsweobtaintheresults
from[HW93],butalsocoversemi-linearautomorphisms(thinkofthisasaF-linear
automorphismcomposedwithafieldautomorphismofF,wherecharF=2).Simi-
larly,wegeneralize[KW92,Proposition5.14],whichdealswithKac-Moodygroups
overalgebraicallyclosedfieldsofcharacteristic0,toarbitraryfieldsofcharacteristic
.2fromtdifferenWebeginbyadaptingsometoolsfrom[HW93].
Lemma2.7.1(Adaptionof[HW93,Lemma2.4,Part2]).Letθbeaquasi-flipofan
RGD-system(G,{Uα}α∈Φ,T)oftype(W,S).Twoθ-stabletwinapartmentsinthe
associatedtwinbuildingcontainingacommonchambercareconjugatebyanelement
ofGθfixingc.
Proof.LetΣandΣbetwoθ-stabletwinapartmentswithnonemptyintersection
Σ∩Σcontainingthechamberc.Thenalsoθ(c)∈Σ∩Σ.TheunipotentradicalU
(cf.Definition1.9.5)oftheBorelsubgroupBstabilizingcactssharplytransitivelyon
thetwinapartmentscontainingc.Hencethereexistsauniqueu∈UmappingΣto
Σ,i.e.,Σ=uΣ.Beingabuildingautomorphismfixingc,ustabilizesthesetΣ∩Σ
chamber-wise.Inparticular,ufixesθ(c),henceu∈U(θ(c))=θ(U),equivalently,
θ(u)∈U.SinceΣ,Σareθ-stable,
uΣ=Σ=θ(Σ)=θ(uΣ)=θ(u)Σ.
Sinceu,θ(u)∈U,andsinceuwasunique,weconcludethatu=θ(u)∈Gθ.
Proposition2.7.2(Adaptionof[HW93,Proposition6.10]).Letθbeaquasi-flipof
anRGD-system(G,{Uα}α∈Φ,T)oftype(W,S)suchthateverychamberiscontained
inaθ-stabletwinapartment.Let{Σi|i∈I}berepresentativesoftheGθ-conjugacy
classesofθ-stabletwinapartmentsinC(resp.θ-stablemaximaltoriinG).IfBis
aBorelgroupoftheRGD-system,then
Gθ\G/B=∼WGθ(Σi)\WG(Σi).
I∈i

38

ositiondecompcosetDouble2.7.

Proof.Assumewearegiventwochambersc,cofthebuildingG/BwhichareGθ-
conjugate,sayc=gcforg∈Gθ.Byourhypotheses,thereisaθ-stabletwin
apartmentΣcontainingc,hencegΣisaθ-stabletwinapartmentcontainingc.
Accordingly,anytwoGθ-conjugatechambersarecontainedinGθ-conjugateθ-stable
twinapartments.NotethatGθ-conjugacyclassesofθ-stabletwinapartmentsarein
one-to-onecorrespondencewiththeGθ-conjugacyclassesofθ-stablemaximaltoriin
.GFurthermore,byLemma2.7.1,iftwoθ-stableapartmentsintersect,thenthey
arealreadyGθ-conjugate.HenceeverychamberliesinauniqueGθ-orbitofθ-stable
apartments,representedbysomeΣi.Therefore,theorbitsofGθonthebuildingG/B,
i.e.,Gθ\G/B,canbeparametrizedviatheΣiandtheGθ-orbitsonthechambersin
.ΣheaciThechambersofeachΣiareinturnparametrizedbyWG(Σi)=Stab(Σi)/Fix(Σi).
TakingtheGθ-actionintoaccount,wecanparametrizetheGθ-conjugacyclassesof
chambersinΣibyWGθ(Σi)\WG(Σi)whichyieldstheclaimeddecomposition.
Weimmediatelyobtainthefollowingcorollary(seealsoCorollaries6.1.4and6.2.2
forapplicationstoalgebraicandKac-Moodygroups):
Corollary2.7.3(ofTheorem2.5.8andProposition2.7.2).Letθbeaquasi-flipof
anRGD-system(G,{Uα}α∈Φ,T)oftype(W,S)whereallrootgroupsareuniquely
2-divisible.ThenwiththenotationfromProposition2.7.2,wehave
Gθ\G/B=∼WGθ(Σi)\WG(Σi).
I∈iCorollary2.7.4(ofCorollary2.7.3andProposition2.6.1).Letθbeaquasi-flip
ofa2-sphericalRGD-system(G,{Uα}α∈Φ,T)oftype(W,S),withnoisolatednodes
inthediagram,andwith2-torsionfreerootgroups.Thenwiththenotationfrom
Proposition2.7.2,wehave
Gθ\G/B=∼WGθ(Σi)\WG(Σi).
I∈iAnalternativeparameterizationofthisdoublecosetdecompositionisgivenin
[HW93],refiningaresultbySpringer[Spr84].In[HW93,Remark6.11]theauthors
sketchhowtoderivethisfromtheparameterizationwegaveabove.Weadaptand
closelyfollowthatremarkinthefollowing.Aspecialcaseofthisoccursagainon
5.2.Equation87,pageProposition2.7.5(Adaptionof[HW93,Proposition6.8]).Letθbeaquasi-flipof
anRGD-system(G,{Uα}α∈Φ,T)oftype(W,S)suchthateverychamberiscontained
inaθ-stabletwinapartment.LetBbeaBorelgroupstabilizingachamberc,letΣ
beaθ-stabletwinapartmentcontainingc.Then
Gθ\G/B=∼WG/Gθ(Σ)={GθgZG(Σ)|g−1θ(g)∈NG(Σ)}.

39

Flips2.

Pringogof.−1θW(ge)∈claimN(Σ)that:evLeteryΣ(Gbθe,aBθ)-stabledoubletwincosetGθapartmenhBhastaconrepresentainingh.ctativ.eBygsatisfy-strong
Gtransitivity,thereexistsg∈GsuchthatΣ=g.Σandh.c=g.c,hencegB=hB.
Thenθ(g.Σ)=θ(g).Σ=g.Σ,i.e.,g−1θ(g)∈NG(Σ).
NowgisuniqueuptorighttranslationbyZG(Σ)andlefttranslationbyGθ.So
ifweputWG/Gθ(Σ)={GθgZG(Σ)|g−1θ(g)−1∈NG(Σ)},thenGθ\G/B=∼WG/Gθ(Σ)
insuchawaythatGθgB↔GθgZG(Σ)ifgθ(g)∈NG(Σ).

40

CHAPTERTHREE

FLIPSINRANK1AND2

Inthischapter,wepresentsomeresultsaboutflipsofMoufangbuildingsofrank1
and2.Theseareofsomeinterestontheirown,butingeneralwillenableustoprove
thingsabouthigher-rankflipsbyreducingtoresultsonlower-rankflips.
Inrank1,thecorrectviewpointistostudyinvolutoryautomorphismsofMoufang
sets,resp.ofrank1groups.Forthis,wefirstfocusourattentiononthemost
basiccase,namelyinvolutoryautomorphismsofSL2(F)andPSL2(F)whereFis
anarbitraryfield.Thiscasesufficestodealwithlocallysplitgroups,suchassplit
algebraicorKac-Moodygroups.ThisisdetailedinSection3.1.
ForflipsofarbitraryMoufangsets,thesituationisnotasgood.Still,wegive
someresultsinSection3.2.Theaimistoshowhowonemightbeable(albeitwith
difficulties)toextendthetheorytogroupsbeyondF-locallysplitones.Asafirst
stepwepresentsomeresultsforSL2(D)whereDisadivisionring.
InSection3.3,westudyinvolutoryautomorphismsofclassicalgeneralizedquad-
rangles,aspecialclassofMoufangbuildingsofrank2(Moufangpolygons).There,
weprimarilyinvestigatewhenthechambersystemofchambersmappedfarawayby
aflipisconnected.KnowingthiswillbecrucialinChapter4.Thisistheresult
ofjointworkbyHendrikVanMaldeghemandtheauthor,see[HVM].Wehopeto
eventuallybeabletotreatarbitraryMoufangquadranglesandhexagons,butthisis
progress.inorkw

3.1.FlipsofSL2(F)andPSL2(F)
WIneexploSectionit3.1.1thatwalleclassifyautomorphismsallinvofolutoryPSL(F)areautomorphismsinducedofbSLy2(F)andautomorphismsPSL2(Fof).
2oSLv2er(F)the,whicfieldshoffollotwwsoandfromthreethefactelementhatts.SL2isAlternativperfectelyif|oneF|≥can4useandtheiseasilyvclassificationerified
ofsufficestoendomorphismsstudyflipsofofSteinSL2b(Ferg).IngroupsSectionorapply3.1.2wtheeresultscomputeinthe[RfixedWW87].pointHencegroupsit

41

3.Flipsinrank1and2

ofglobaltheseflips.transitivityThispropworkertieswillofbeflipsusedfromintheirChapterlocal5(rankwhere1)wbeehastrivvior.etodetermine

3.1.1.ClassifyingflipsofSL2(F)
InordertobeabletounderstandinvolutoryautomorphismsofG:=SL2(F),consider
SL2(F)asamatrixgroupactingonitsnaturalmodule.LetTdenotethesubgroup
ofdiagonalmatrices,whichisamaximaltorusofG.LetU+resp.U−denotethe
subgroupsofstrictlyupperresp.lowertriangularunipotentmatrices,whichareroot
subgroupswithrespecttotherootsystemoftypeA1associatedtoT.Thestandard
BorelsubgroupsofGthenarethegroupsB+:=T.U+andB−:=T.U−(thesub-
groupsofallupperresp.lowermatrices).Finally,settingN:=NG(T)weobtaina
(twin)-BN-pair(B+,B−,N)asdefinedinSection1.8.
Recallthattwoautomorphismsφ,ψofGareconjugateiftheyareconjugate
withinAut(G),thatis,thereexistsω∈Aut(G)suchthatφ=ω◦ψ◦ω−1.For
A∈G,denotebyIntAtheinnerautomorphismx→AxA−1.Byslightabuseof
notation,wealsousethisnotationifA∈GL2(F).
Lemma3.1.1.EveryinvolutoryautomorphismofG=SL2(F)isconjugatetoan
involutoryautomorphismwhichinterchangesB+andB−.
Proof.TheautomorphismsofSL2(F)aredeterminedin[SW28](agapintheproof
giventhereisclosedin[Hua48]):AnyautomorphismθofGisobtainedbycomposing
afieldautomorphismσwithIntg,whereg∈GL2(F).Onereadilyobservesthatifθ
hasorderσ2,σhasorderatmost∗2.Fσurthermore,fromθ2(x)=Intggσ(x)=xfollows
thatgg=λIforsomeλ∈F(i.e.,ggisanelementofthecenterofGL2(F)).
Finally,sinceσmapsB+toitself,θmapsanyconjugateofB+againtoaconjugate.
Assumethatforallh∈G,θmapsthegrouphB+h−1toitself,equivalently,maps
hB−h−1toitself.Weclaimthatθthenistheidentity:Forh∈Gandε∈{+,−},we
haveθ(hBεh−1)=hBεh−1whichimpliesh−1ghσ∈NG(B+)∩NG(B−)=B+∩B−=
T.Settingh=I,weconcludeg∈T,sayg=0tt−01witht∈F∗.Setting,h=(0111),
fromh−1ghσ=0ttt−−t1−1∈T,wededucet=t−1,henceg=tI.Thus,forallh∈G
wehaveh−1hσ∈T.Settinghx=(011x)withx∈Farbitrary,h−1hσ=(01xσ1−x)∈T
impliesσ=idF,thusθisindeedtheidentitymap.
Henceifθisaninvolution,thereexistsa∈GsuchthataB+a−1isdistinctfrom
itsimageunderθ.Thenθ:=Inta◦θ◦(Inta)−1isaninvolutionconjugatetoθ
whichmapsB+toaconjugateyB+y−1differentfromB+.Thereexistsaunique
u∈U+suchthatyB+y−1=u−1B−u.Thenθ:=Intu◦θ◦(Intu)−1isaninvolution
conjugatetoθwhichinterchangesB+andB−.
Lemma3.1.2.AnyinvolutoryautomorphismθofGwhichinterchangesB+and
B−isoftheform
θ:X→(δ001)Xσ10δ0−1,
whereσisafieldautomorphismoforderatmost2,andδ∈FixF∗(σ).

42

3.1.FlipsofSL2(F)andPSL2(F)

Proof.Weknowthatθ=Intg◦σforsomeσ∈Aut(F)oforderatmost2,andsome
g∈GL2(F).SinceθinterchangesB+andB−,itfollowsthatθstabilizesT=B+∩B−
andinterchangesU+andU−.AsσstabilizesU+andU−,weconcludethatIntgmust
interchangeU+andU−.Onereadilycomputesthattheng=(c00b)forsomeb,c∈F∗.
Moreover,θ2=Idimpliesggσ=λIforsomeλ∈F∗.Butggσ=(bc0σbσ0c).Hence
λ=bcσ=bσc∈FixF∗(σ).Settingδ:=bc=bbλσ∈FixF∗(σ),weobservethatgand
(δ001)inducethesameautomorphism.
Theprecedinglemmamotivatesthefollowingdefinition:
Definition3.1.3.Forσ∈Aut(F)oforderatmost2andδ∈FixF∗(σ)wedefine
olutionvinstandardtheθδ,σ:SL2(F)→SL2(F):X→θδ,σ(X)=xδXσxδ−1,
wherexδ:=(δ001).Equivalently,θδ,σ=Intxδ◦σ.
Byslightabuseofnotation,wewillusethesamesymbolθδ,σtodenotetheinduced
fliponPSL2(F).Altogether,wehaveprovedthefollowinginthissection:
Proposition3.1.4.ForeveryinvolutoryautomorphismofSL2(F)orPSL2(F)there
existσ∈Aut(F)oforderatmost2andδ∈FixF∗(σ)suchthatθisconjugatetoθδ,σ.
Wenowdescribewhentwostandardinvolutionsareconjugate.
Proposition3.1.5.Twostandardinvolutionsθδ,σandθε,τareconjugateifand
onlyifthereexistsρ∈Aut(F)suchthatσ=ρτρ−1andδ/ερ∈Nσ(F∗),where
Nσ(x):=xxσ.
Proof.Ifθδ,σandθε,τareconjugate,thereexistsanautomorphismφ=Intg◦ρ,with
ρ∈Aut(F)andg∈GL2(F),suchthatourtwostandardinvolutionsareconjugate
byφ.Thatis,
θδ,σ◦φ=φ◦θε,τ
⇐⇒(Intxδ◦σ)◦(Intg◦ρ)=(Intg◦ρ)◦(Intxε◦τ)
⇐⇒Intxδ◦Intgσ◦σ◦ρ=Intg◦Intxερ◦ρ◦τ
⇐⇒Intxδ◦Intgσ◦σ=Intg◦Intxερ◦(ρτρ−1),
wherexδ,xεasinDefinition3.1.3.Hencewemusthaveσ=ρτρ−1(toseethis,
notethatAut(G)isthesemi-directproductofthenormalsubgroupof“inner”
automorphismsinducedbyGL2(F),andthefieldautomorphisms).Accordingly,
Intxδ◦Intgσ=Intg◦Intxερ,whichimpliesxδgσ=λgxερforsomeλ∈F∗.Setting
g=(cadb),thisisequivalentto
cσdσερba
δaσδbσ=λερdc,

43

3.Flipsinrank1and2

fromwhichwededucebycomparingcoefficientsthatεδρ=Nσ(λ).
Conversely,supposethereexistρ∈Aut(F)andλ∈F∗suchthatσ=ρτρ−1and
δ/ερ=Nσ(λ).Setg:=(01λ0)andφ=Intg◦ρ.Thenθδ,σ=φ◦θε,τ◦φ−1.
Corollary3.1.6.LetInv(F)denoteasetofrepresentativesoftheconjugacyclasses
ofautomorphismsofFoforderatmost2.Thentheconjugacyclassesofinvolutory
automorphismsofGcorrespondone-to-onetothedisjointunion
FixF∗(σ)/Nσ(F∗),whereNσ(x):=xxσ.
σ∈Inv(F)
3.1.2.Centralizersofflips
Wenowturnourattentiontothecentralizersofagivenflipθ,whichwillbeof
interestinChapter5.Wecompute
uv01uσvσ0δ−1uv
CSL2(F)(θδ,σ)=wxδ0wσxσ10=wx,ux−vw=1.
Itisnoweasytoverifythat
vuKδ,σ:=CSL2(F)(θδ,σ)=δvσuσ|uuσ−δvvσ=1,
whichispreciselythegrouppreservingtheσ-sesquilinearform
fδ,σ(x,y):=xT−0δ10yσ,x,y∈F2,
onthevectorspaceF2anditsassociatedσ-quadraticformqδ,σgivenbyqδ,σ(x):=
fδ,σ(x,x).Thisalternativecharacterizationwillturnouttobequiteuseful.
ForPSL2(F),thesituationisslightlydifferent.DenotebyZthecenterofSL2(F),
soPSL2(F)=SL2(F)/Z,accordinglythecentralizerofθinPSL2(F)isCPSL2(F)(θ)=
{gZ∈PSL2(F)|(gZ)θ=gZ}.Wearemainlyinterestedintheactionofthis
centralizeronP1(F).SincetheactionofPSL2(F)isinducedbythatofSL2(F),this
boilsdowntostudyingthepreimageofthecentralizerinSL2(F),whichsuggeststhe
definition:wingfolloDefinition3.1.7.LetθbeanautomorphismofSL2(F).Wedefinetheprojective
centralizerofθinSL2(F)asthegroupPCSL2(F)(θ):={g∈SL2(F)|gθ∈gZ},
whichisthepreimageofCPSL2(θ)inSL2(F)underthecanonicalprojectionπ:SL2→
.PSL2Wecompute
PKδ,σ:=PCSL2(F)(θδ,σ)=δvεuσuεvσ|uuσ−δvvσ=ε,ε∈{+1,−1}.
WhileKδ,σpreservestheσ-sesquilinearformfδ,σ(x,y)anditsassociatedσ-quadratic
formqδ,σ(x),thegroupPKδ,σpreservestheseformsonlyuptosign.

44

setsMoufangofFlips3.2.

setsMoufangofFlips3.2.Wenowturnourattentiontothegeneralrank1case,i.e.,thestudyofflipsof
Moufangsets.TheresultspresentedhereareduetoTomDeMedts,cf.[DMGH09,
Section5].WecloselyfollowthenotationintroducedinSection1.10.
Thegoalofthissectionistocharacterizetheinvolutionsθ∈Aut(G)interchanging
U∞andU0.Suchaninvolutionθmapseachαatosomeγaϕandeachγbtosome
αbψ.Sinceθ∈Aut(G),wehaveϕ,ψ∈Aut(U).Moreover,θ2=idimpliesψ=ϕ−1.
Inparticular,θiscompetelydeterminedbyϕ.Moreprecisely,foreachϕ∈Aut(U),
defineewαa→γaϕ
θϕ:U∞∪U0→U0∪U∞:γa→αaϕ−1.
ThequestioniswhenθϕextendstoanautomorphismofG.Observethatifθϕ
extends,thenthisextensionisuniqueandisinvolutory,sinceθisinvolutoryon
U∞∪U0andG=U∞∪U0.
Proposition3.2.1.Letϕ∈Aut(U).Thenθϕextendstoan(involutory)automor-
phismofGifandonlyif(ϕτ)2=id.Moreover,ifthisisthecase,thenϕ∈Aut(M).
Proof.Letθ:=θϕandβ:=ϕτ.Assumefirstthatθextendstoanautomorphismχ
Then.Gofχ(Ua)=χ(U0αa)=χ(U0)χ(αa)=U∞γaϕ=Uaϕτ=Uaβ(3.1)
foralla∈U.Sinceθ2istheidentityonU∞∪U0andsinceG=U∞,U0,this
impliesthatχ2=1andhenceβ2=1.
Conversely,assumethatβ2=1,andletχβbeasinDefinition1.10.7.Thenfor
,Uaall∈χβ(αa)=αaϕτ=ατaϕ=γaϕ,
χβ(γa)=γaϕτ=γaτ−1ϕ−1=αaϕ−1=αaϕ−1;
henceχβandθcoincideonU∞∪U0.Notethatχβisan(inner)automorphismof
Sym(X),andhencethesamecalculationasinequation(3.1)(withχβinplaceofχ)
showsthatβ∈Aut(M).HencetherestrictionofχβtoGisanautomorphismofG;
thisisthe(unique)extensionofθtoanelementofAut(G).
Finally,sincewehavejustshownthatβ∈Aut(M)andsinceobviouslyτ∈
Aut(M),weconcludethatϕ∈Aut(M)aswell.
Definition3.2.2.Anautomorphismϕ∈Aut(U)withthepropertythat(ϕτ)2=1
willbecalledaflipautomorphismofM.
Thefollowingtheoremgivesimportantinformationaboutsuchflipautomorphisms.
InaccordancewiththeusualconventionsemployedintheoryofMoufangsetsand
withournotationfromSection1.10,wewillalwaysdenotetheactionofapermuta-
tiononasetontheright,i.e.,wewillwriteaϕratherthanϕ(a).

45

3.Flipsinrank1and2

Theorem3.2.3.LetMbeaMoufangset,andletϕbeaflipautomorphismofM.
Thengaϕ=ϕ∙ha∙ϕ
foralla∈U∗.Moreover,ifeisanidentityelementofM,i.e.,τ=µ−e,then
ϕ∈Str(M,e)∩Aut(M).
Proof.Foreacha∈U∗,themapgaistheHuamapofawithτreplacedbyτ−1,
andhencegaϕ=τ−1αaϕτα−aϕττ−1α−(−(−aϕτ))τ−1foralla∈U∗.Usingthefacts
thatαaϕ=αaϕ,ϕτ=τ−1ϕ−1and(−a)ϕ=−aϕseveraltimes,wegetϕ−1gaϕ=
ταaτ−1α−aτ−1τα−(−(−aτ−1))τϕ=haϕ.Inparticular,ifeisanidentityelementofM,
thenhe=1andhenceϕ−1geϕ=ϕ.Itfollowsthatϕgeϕ−1gaϕ=haϕforalla∈U∗.
However,geϕ−1gaϕ=(µeµeϕ)−1(µeµaϕ)=(µ−eµeϕ)−1(µ−eµaϕ)=h−eϕ1haϕ=h(aϕeϕ)and
hencehaϕ=ϕh(aϕeϕ)foralla∈U∗,provingthatϕ∈Str(M,e).Thefactthat
ϕ∈Aut(M)wasshowninProposition3.2.1above.
WewillnowillustratethestrengthofTheorem3.2.3byexplicitlydeterminingall
flipsofPSL2(D),whereDisafieldoraskewfield.
Proposition3.2.4.LetDbeanarbitraryfieldorskewfield,andletM=M(D)be
thecorresp∗onding∗Moufang−1set,i.e.,theMoufangsetM=M(U,τ)whereU:=(D,+)
andτ:D→D:x→−x.
(i)LetϕbeaflipautomorphismofM.Thenthereexistsanautomorphismor
anti-automorphismσofDandanelementε∈FixD(σ)suchthatxϕ=εσ(x)
forallx∈D.Ifσisanautomorphism,thenσ2(x)=ε−1xεforallx∈D;ifσ
isananti-automorphism,thenσ2=1.
(ii)Conversely,supposethateitherσisananti-automorphismoforder2andε∈
FixD(σ)isarbitrary,orσisanautomorphismsuchthatσ2(x)=ε−1xεforsome
ε∈FixD(σ).Thenthemapϕ:D→D:x→εσ(x)isaflipautomorphismof
.MProof.(i)Observethat1∈D∗isanidentityelementofM;alsonotethatτ2=id.
Foralla,b∈U∗,wehavebha=aba.Thecondition(ϕτ)2=1translatesto
(a−1)ϕ=(aϕ−1)−1(3.2)
foralla∈D∗.Letε:=1ϕ;thenbha(1ϕ)=bh1−ϕ1ha=aε−1bε−1aforalla,b∈U∗.
ByTheorem3.2.3,ϕ∈Str(M,e),whichmeansthatbhaϕ=bϕh(1ϕ)forall
a,b∈U∗,orexplicitly,(aba)ϕ=aϕ∙ε−1∙bϕ∙ε−1∙aϕforallaϕa,b∈D∗.
Nowletσ(a):=ε−1∙aϕforalla∈D.Thenσ∈Aut(U),andtheprevious
equationcanberewrittenasσ(aba)=σ(a)σ(b)σ(a)foralla,b∈D,i.e.,σ
isaJordanautomorphismofD.ItisawellknownresultbyJacobsonand
Rickart[JR50](seealso[Jac68,page2]),whichsimplyamountstocalculating
thatσ(ab)−σ(a)σ(b)∙σ(ab)−σ(b)σ(a)=0,thatσiseitheranauto-
morphismorananti-automorphismofD.Nowbyequation(3.2),wehave

46

3.3.Classicalquadrangles

(ε−1)ϕ=(εϕ−1)−1=1−1=1,andhenceσ(ε−1)=ε−1;sinceσisanautomor-
phismoranti-automorphism,itfollowsthatσ(ε)=ε.Finally,againbyequa-
tion(3.2),weobtainσ(εσ(a))=σ(aϕ)=σ((a−1)ϕ−1)−1=σ((a−1)ϕ−1)−1=
(ε−1a−1)−1=aεforalla∈D∗.Ifσisanautomorphism,thenthiscanbe
rewrittenasεσ2(a)=aε,orσ2(a)=ε−1aε;ifσisananti-automorphism,we
getσ2(a)ε=aε,i.e.,σ2=1.
(ii)Itsufficestocheckthatequation(3.2)holds.Thisamountstocheckingthat
εσ(a−1)=(σ−1(ε−1a))−1foralla∈D.Itisstraightforwardtocheckthatthis
isvalidinbothcases.
By[RWW87]theflipsofSL2(D)arejusttheliftsoftheflipsofPSL2(D).

uadranglesqsicalClas3.3.Inthissectionwestudyinvolutoryautomorphismsofclassicalquadrangles.First,
wegivesomeauxiliaryresultsonMoufangsetsinSection3.3.1.
InSection3.3.2wegiveanalgebraicdescriptionofclassicalquadrangleswhichwe
usethroughouttherestofthissection.Ourmainreferenceforclassicalquadrangles
2.3].Section[VM98,isForanyinvolutoryautomorphismθofageneralizedquadrangle,wedefinetheflip-
flopsystemRθasthesetofallchamberscforwhichδ(c,θ(c))ismaximalamong
allchambers(hereδistheWeylmetriconthequadrangle).Thisisaspecialcase
ofamoregeneraldefinitioninChapter4.Wewillshowthatwhenourquadrangle
isdefinedoverafieldofcharacteristicdifferentfrom2,theflip-flopsystemRθis
connectedregardlessofthechoiceofθ.
Note:Weonlydealwithclassicalquadranglesdefinedovercommutativefields
incharacteristicdifferentfrom2.Withsomeeffortitshouldbepossibletorefine
theargumentstoworkoverskewfields,and(atleasttheconnectednessresult)in
characteristic2.However,theargumentsbecomealotmoreinvolved.Sinceweare
primarilyinterestedinthesplitcase,wedecidednottotrytoachievefullgenerality
here.

3.3.1.SomeauxiliaryresultsonMoufangsets
WereferthereadertoSection1.10forthebasicsaboutMoufangsets.Here,weonly
presentsomenon-standardextensionstothetheory,whichwhilebasicandrelatively
simple,mostlyseemtonotbeintheliterature.
Forconvenience,wewillwriteM(X,U)asashorthandfor(X,(Ux)x∈X).
Definition3.3.1.A(proper)MoufangsubsetofaMoufangsetM(X,U)isa
MoufangsetM(Y,V)suchthatYisa(proper)subsetofXandforally∈Ywe
haveVy≤Uy.WealsowriteM(Y,V)≤M(X,U)(resp.M(Y,V)<M(X,U)ifYis
aproperMoufangsubset).

47

3.Flipsinrank1and2

Weextendthisnotionslightly:
Definition3.3.2.AgeneralizedMoufangsubsetofaMoufangsetM(X,U)is
subsetYofXsuchthatif|Y|≥2,thenthereexistsaMoufangsubsetM(Y,V)of
M(X,U)onV.WealsowriteY≤M(X,U)(andY<M(X,U)ifYisaproper
).XofsubsetRemark3.3.3.Arelatedconceptisthatofarootsubgroup,see[Seg08,Section3].
Awell-knownfactisthat(generalized)Moufangsubsetsoccurnaturallyasfixed
pointsetsofautomorphismsofMoufangsets:
Lemma3.3.4.LetM(X,U)beaMoufangset,letσbeanautomorphismofM(X,U).
DenotebyYthesetoffixedpointsofσ.ThenYisageneralizedMoufangsubsetof
M(X,U).
Proof.If|Y|<2thereisnothingtoshow.Sosuppose|Y|≥2.Foreveryy∈Y,
chooseanelementyinYdifferentfromyandsetVy:={g∈Uy|y.g∈Y}.We
needtoverifythatforeachy∈YthesetVyiswell-defined(i.e.,independentofthe
choiceofy),isasubgroupofUyandactssharplytransitivelyonY\{y}.
Sofirstobservethatforeachy∈Y,wehavey.Vy=Y\{y},duetothewaywe
definedVyandsinceUyactssharplytransitivelyonX\{y}.Thenforallg∈Vywe
evhay.gσ=y.(σ−1gσ)=y.gσ=y.g
andthusbyregularity,g=gσ.
ToseethatVymapsYtoY(andhenceformsagroup),assumetheexistence
ofsomeg∈Vyandsomey∈Ysuchthaty.g=:z∈/Y.Butthisyieldsa
tradiction:conz=y.g=y.(σ−1gσ)=y.gσ=z.σ=z.
HenceVyistheuniquesubgroupofUyactingsharplytransitivelyonY\{y}foreach
y∈Y.ItfollowsthatVyisindependentofthechoiceofy,andpermutestheset
{Vx|x∈Y}byconjugation.
TheintersectionoftwogeneralizedMoufangsubsetsofagivenMoufangsetis
againageneralizedMoufangsubset.
Lemma3.3.5.LetM(X,U)beaMoufangset,andletYandZbetwogeneralized
Moufangsubsets.ThenY∩ZisageneralizedMoufangsubset.
Proof.SetA:=Y∩Z.If|A|<2,thereisnothingtoshow.SoassumeAcontains
atleasttwodistinctelements0and∞.ThenYandZarethebasesetsoftwo
MoufangsubsetsM(Y,V)andM(Z,W)ofM(X,U).SetB∞:=V∞∩W∞.SinceX
isaMoufangset,thereexistsforeacha∈Aauniqueelementg∈U∞whichmaps
0toa.ThereforegmustalsobecontainedinV∞andW∞andhenceinB∞.Thus
B∞actsregularlyonA\{∞}.Also,gpermutestheBxbyconjugation,forwehave
Bxg=(Vx∩Wx)g=Vxg∩Wxg=Vxg∩Wxg=Bxg.
Since0and∞werearbitrary,thiscompletestheproof.

48

3.3.Classicalquadrangles

NextweobservethatMoufangsubsetscanbeatmostabout“halfasbig”asthe
Moufangsettheyarecontainedin.Inparticular,theorderoftherootsubgroupsVx
mustdividetheorderoftheoriginalrootgroupsUx,andhencehaveatleastindex
2.Thefollowinglemmamakesthisprecise.
Lemma3.3.6.LetM(X,U)beaMoufangsetandYapropergeneralizedMoufang
subset.Thenthefollowingholds:
(1)|Y|≤|X\Y|+1.
(2)IfXisinfinitethenX\Ycannotbefinite.
(3)IfXisfinite,denotebypthesmallestprimedividing|X|−1.Then
pp(|Y|−1)≤|X|−1≤p−1|X\Y|.
(4)IfXisfinitethen2|Y|≤|X|+1and|X|≤2|X\Y|+1.
Proof.(1)Theclaimistrivialif|Y|≤1.Soassumew.l.o.g.thatYcontainstwo
distinctelements0and∞.SinceYX,weknowinfactthatV0<U0.Thus
|U0:V0|≥2,whichimpliesthat|V0|≤|U0\V0|.ButU0\V0isinnatural
bijectionwithX\Y(identifyeachelementginthefirstsetwith0g).Likewise,
Y\{∞}isinnaturalbijectionwithV0.
(2)IfXwasinfiniteandX\Yfinite,thenYwouldbeinfinite,contradicting(1).
(3)IfXisfinite,then|X|=|U0|+1and|Y|=|V0|+1,thuspisthesmallest
primedividing|U0|,andso|U0:V0|≥p.Forthisreasonp(|Y|−1)≤|X|−1.
Byadding(p−1)|X|tobothsides,anddividingbyp−1,wegetthesecond
.yinequalit(4)Followsfrom(3)byusingthat2≤p.
Asaconsequence,aMoufangsetcannotbetheunionoftwoofitspropergener-
alizedMoufangsubsets,unlessitisverysmall.Thisisstilltrueifweallowadding
oneextrapointtotheunion.
Lemma3.3.7.LetM(X,U)beMoufangset,andletYandZbetwopropergen-
eralizedMoufangsubsets.ThenifX=Y∪Z∪{a}forsomea∈X,wehave
|X|≤5.
Proof.SupposeX=Y∪Z∪{a}and|X|>5.
ByLemma3.3.6,ifXisinfinitethenbothYandZandtheircomplementsmust
beinfinite.IfXisfinite,then5<|X|≤2|X\Y|+1impliesthat|X\Y|>2.In
eithercase,bothX\Y⊆Z\Y∪{a}andX\Z⊆Y\Z∪{a}eachcontainatleast
threeelements.HenceZ\Y,Y\Z,YandZallcontainatleasttwoelementseach.
Inparticular,YandZformMoufangsubsetsM(Y,V)andM(Z,W).

49

3.Flipsinrank1and2

Fixtwodistinctelements0and∞inY\Zandchooseanyg∈U∞whichmaps0
intoZ\Y.Clearly,g∈/V∞andinfactgmapsY\{∞}intoasubsetofZ\Y∪{a}
(becauseU∞actsregularlyonX\{∞},andV∞<U∞alreadyactsregularlyon
Y\{∞}).NowYgisagainaMoufangsetwithrootgroupsVyg.Hence−b1yLemma
3.3.5,Z:=Z∩YgisageneralizedMoufangsubset.ButthenY:=Zgisalsoa
.subsetMoufanggeneralizedIfa∈Y∪Z,thenY=Y\{∞}andbyLemma3.3.6,weobtain|Y|≤2|Y\Y|+1≤
3andbysymmetry|Z|≤3,thus|X|≤6.Howeverif|X|=6=5+1,theonly
Moufangsubsetsareofsize2and1,soaMoufangsetofsize6cannotbecoveredby
twoMoufangsubsetsandasinglepoint.
Ifa∈/Y∪Z,thenwemayhaveY=Y\{ag−1,∞}.AgainbyLemma3.3.6,we
obtain|Y|≤2|Y\Y|+1≤5.Bysymmetryalso|Z|≤5hence|X|≤11.
TheremainingpossibilitiescanbeexcludedviatheclassificationoffiniteMoufang
sets(see[HKS72]and[Shu72]),orviadirectcomputations(usingasimplecomputer
program).Brieflysketched,theargumentsareasfollows:ForaMoufangsetof
sizen+1,anysubsethastohavesizem+1withmdividingn.For6=5+1,
8=7+1and10=9+1,ithenceisclearthattwosubsetsplusonepointcannot
covereverything.Moreover,theMoufangsetsofsize7and11aresharplytransitive,
andhavenonontrivialMoufangsubsets.Thecasewhere|X|=9=8+1isthe
hardesttoexclude,asitcouldpotentiallyhaveasubsetofsize5=4+1,butnone
ofthethreeMoufangsetsofsize9hasaMoufangsubsetofsize5.1

ettingsCommon3.3.2.Wefollowpreciselythesettingin[VM98,Section2.3.1],andwillomitsomedetails
giventhere.Thereadermayhencewishtoconsultloc.cit.paralleltoreadingthis
section.LetKbeaskewfield,charK=2,andσananti-automorphismoforderatmost
2(thusifσ=id,thenKiscommutative).LetVbea–notnecessarilyfinite-
dimensional–rightvectorspaceoverKandletg:V×V→Kbea(σ,1)-linear
form.Wedefinef:V×V→Kasfollows:
f(x,y)=g(x,y)+g(y,x).
Thenfisa(σ-)Hermitianform.DenoteKσ:={tσ−t|t∈K}.Wedefine
q:V→K/Kσas
q(x)=g(x,x)+Kσ,
forallx∈V.Thenqisaσ-quadraticform.
Assumenowthatqisnon-degenerateandhasWittindex2.Weobtainaclassical
generalizedquadrangleΓbytakingthetotallyisotropic1-spacesaspoints,andthe
1Alternatively,onecanusethatfor|X|=9and|Y|=|Z|=5,a∈/Y∪Zimplies|Y∩Z|≥2.
WButcothenverUonecanexceptchooseforonedistinctelemepnoint.tsBut0,∞thenin|YU∩|Z≤,4.andgetsthattherootgroupsV∞and
∞∞∞

50

3.3.Classicalquadrangles

totallyisotropic2-spacesaslines.Onecanshowthattwopointsvandware
incidentifandonlyiff(v,w)=0.

Proposition3.3.8.Letqbeanon-degenerateσ-quadraticformasabove,ofWitt
index2,withcorrespondinggeneralizedquadrangleΓ.LetθbeacollineationofΓof
order2.Letp1,p2,p3,p4befourpointsofΓspanningaθ-stablethinsubquadrangle,
wherep1isoppositep4,andp2isoppositep3.Thenthefollowinghold:
(1)ThereexistsavectorsubspaceV0ofV,adirectsumdecomposition
V=e1Ke2Ke3Ke4KV0
andanon-degenerateanisotropicσ-quadraticformq0:V0→K/Kσ,suchthat
forallv=e1x1+e2x2+e3x3+e4x4+v0withxi∈K,i∈{1,2,3,4}and
v0∈V0,
q(v)=x1σx4+x2σx3+q0(v0).
Moreover,pi=eifori∈{1,2,3,4},andf(ei,ej)equals0unless{i,j}=
{1,4}or{i,j}={2,3},inwhichcaseitequals1.
(2)DenoteV:=e1,e2,e3,e4.ThereexistA∈GL(V),D∈GL(V0),γ∈Aut(K)
oforderatmost2andλ∈C(K)∗,suchthatforallv∈V,v0∈V0,
θ(v+v0)=Avγ+Dv0γ,
andAAγ=λI,DDγ=λI.
Proof.(1)Thisis[VM98,Proposition2.3.4].Inparticular,
V0:={v∈V|f(v,ei)=0,fori=1,2,3,4}.
Bychoosingappropriatescalarmultiplesoftheei,wecanensurethecondition
onf(ei,ej).
(2)ByProposition4.6.5inloc.cit.,θisinducedbyaprojectivesemilineartrans-
formationoftheunderlyingvectorspaceV.HencethereisT∈GL(V)and
γ∈Aut(K)suchthatθ(v)=Tvγforallisotropicvectorsv.Sinceθhas
order2,γhasatmostorder2,andthereexistsλ∈C(K)∗suchthatTTγ=λI.
Byslightabuseofnotation,wealsouseθtodenotethesemilinearmapv→Tvγ
.VonNowVisθ-stablebyhypothesis,andorthogonaltoV0bydefinitionofthe
latter.Sinceθisacollineation,V0mustalsobeθ-stable.Thereforewecan
block-decomposeTasstated.

51

3.Flipsinrank1and2

tscendeDirect3.3.3.ThefollowingresultswillbeusefulinChapter4toprovetheso-called“directdescent”
property.Wegivethemherebecausetheyalsoyieldconnectednessoftheflip-flop
systemifnoPhanchambersexists,andingeneraltheirproofsperfectlyfitinwith
section.thisofresttheProposition3.3.9.ConsideraclassicalquadrangleoveraskewfieldK,inwhich
alineLexistssuchthatLisoppositetoθ(L),andsuchthatallpointsponLare
collineartoθ(p).ThenKiscommutative.IfcharK=2ordim(V0)=0,thenevery
pointiscollineartoitsimage.
Proof.Taketwoarbitrarypointsp1,p2onL.Thenp1,p2,θ(p1),θ(p2)formaθ-stable
thinquadrangle.ThenbyProposition3.3.8,thereareisotropicvectorse1,e2,e3,e4
suchthatp1=e1,θ(p1)=e2,θ(p2)=e4,p2=e3,and
θ(v+v0)=Avγ+Dv0γ,
whereA∈GL(V),D∈GL(V0),γ∈Aut(K)oforderatmost2andλ∈C(K)∗.In
particular,sinceθswapse1withe2,ande3withe4,wehave
bA=ad,a,b,c,d∈K∗.
cSinceθisonlydetermineduptoascalarfactor,wemaychoosea=1;fromAAγ=λI,
wededuceλ=b=bγ=dcγ=cdγ,henceλγ=λ.
Byhypothesis,everypointp∈L=e1,e3iscollineartoitsimage.Soforall
:Kνµ,∈0=f(e1ν+e3µ,θ(e1ν+e3µ))=f(e1ν+e3µ,e2νγ+e4cµγ)
γγγγ=f(e1ν,e2ν)+f(e3µ,e2ν)+f(e1ν,e4cµ)+f(e3µ,e4cµ)
=µσνγ+νσcµγ.
Settingµ=ν=1wefindc=−1,henced=−λ.Forν=1andµarbitrary,
weobtainγ=σ.Sinceγisanautomorphism,butσananti-automorphism,we
concludethatKiscommutative.Allinall,weget
θ(e1v1+e2v2+e3v3+e4v4+v0)=e1λv2σ+e2v1σ−e3λv4σ−e4v3σ+Dv0σ.(3.3)
Showingthateverypointofthequadrangleiscollineartoitsimageisequivalentto
showingthatforeveryisotropicvectorvwithθ(v)=v,wehavef(θ(v),v)=0.
Letv=e1v1+e2v2+e3v3+e4v4+v0∈Varbitrary.Wecompute
f(θ(v),v)=f(e1λv2σ+e2v1σ−e3λv4σ−e4v3σ+Dv0σ,e1v1+e2v2+e3v3+e4v4+v0)
=f(e1λv2σ,e4v4)+f(e2v1σ,e3v3)+f(−e3λv4σ,e2v2)+f(−e4v3σ,e1v1)
+f(Dv0σ,v0)
=v2λv4−v3v1+v1v3−v4λv2+f(Dv0σ,v0)
=f(Dvσ0,v0).

52

3.3.Classicalquadrangles

Toprovethatf(Dv0σ,v0)iszeroforallv0∈V0,choosex∈Ksuchthatx≡q0(v0)
(modKσ).Setv:=e1x−e4+v0;thisisanisotropicvector.Similarly,choosey∈K
suchthaty≡q0(Dv0σ)(modKσ)andsetw:=e1f(v0,Dv0σ)+e2y−e3+Dv0σ.This
isalsoanisotropicvector,andviscollineartow,since
f(v,w)=f(−e4,e1f(v0,Dv0σ))+f(v0,Dv0σ)=0.
Sinceθmapslinestolines,itfollowsthat
0=f(θ(v),θ(w))=f(e2xσ+e3λ+Dv0σ,e1λyσ+e2f(v0,Dv0σ)σ+e4+θ(Dv0σ))
=λf(v0,Dv0σ)σ+f(Dv0σ,λv0)
=2λf(Dv0σ,v0).
HenceifcharK=2,weindeedhavef(Dv0σ,v0)=0andthereforeasclaimed,
f(θ(v),v)=0forallv∈V.
Remark3.3.10.IfcharK=2onecanconstructexampleswhichotherwisesatisfy
theassumptionsofProposition3.3.9butwherepointsexistthataremappedto
oppositeones.ForexamplechooseK=F4,dim(V0)=1andγ=σequaltothe
Frobeniusautomorphism.Considertheautomorphisminequation(3.3)withD=
λ=1.Thenthepointe1+e4α+v0isnotcollineartoitsimagee2+e3α+v0,
whereα∈F4\F2andv0∈V0\{0}.
Proposition3.3.11.ConsideraclassicalquadrangleoveraskewfieldKinwhicha
pointpexistssuchthatpisoppositetoitsimageθ(p),andsuchthatalllinesthrough
pcontainafixedpoint.Ifdim(V0)>0,orifcharK=2andKiscommutative,then
everylinecontainsafixedpoint.
Proof.TaketwoarbitrarylinesM,Nthroughp.DenotebyqMresp.qNthe(by
ourhypothesisthatpisoppositeθ(p)unique)fixedpointsonMresp.N.Then
p,qM,θ(p),qNformaθ-stablethinquadrangle.ByProposition3.3.8,thereare
isotropicvectorse1,e2,e3,e4suchthatp=e1,qM=e2,θ(p)=e4,qN=e3,
andwehave
θ(v+v0)=Avγ+Dv0γ,
whereA∈GL(V),D∈GL(V0),γ∈Aut(K)oforderatmost2andλ∈C(K)∗.
Inparticular,sinceθfixesqM=e2andqN=e3butinterchangesp=e1and
θ(p)=e4,d
A=bc,a,b,c,d∈K∗.
aSinceθisonlydetermineduptoascalarfactor,choosea=1;fromAAγ=λI,we
deduceλ=dγ=bbγ=ccγ=d,henceλγ=λ.
Byhypothesis,alllinesthroughe1containafixedpoint.Inotherwords,forany
isotropicvectorvcollineartobutdifferentfrome1,thereexistsascalarαvsuchthat
e1αv+visafixedpoint:
e1αv+v=θ(e1αv+v)=e4αvγ+θ(v).

53

3.Flipsinrank1and2

Consequently,thereexistsβv∈K∗suchthat
γγ(e1αv+v)βv=e4αv+θ(v)⇐⇒θ(v)−vβv=e1αvβv−e4αv∈e1,e4.(3.4)
WenowmakeacasedistinctionbasedonV0.
V0=0:WestartbyshowingthatDisamultipleoftheidentitymatrix.Letv0∈V0
bearbitrary,letx∈Kbesothatq(v0)≡x(modKσ).Thenv:=e2x−e3+v0
isanontrivialisotropicvector,collineartobutdifferentfrome1.Inparticular,
thevalueβvasaboveisdefined.Wecompute:
θ(v)−vβv=θ(e2x−e3+v0)−(e2x−e3+v0)βv
=(e2bxγ−e3c+Dv0γ)−e2xβv+e3βv−v0βv
=e2(bxγ−xβv)+e3(βv−c)+(Dv0γ−v0βv).
ButbyEquation3.4wehaveθ(v)−vβv∈e1,e4.Consequently,bycomparing
coefficients,βv=c,andsincev0wasarbitrary,D=cIandγ:x→c−1xc.
From(bxγ−xβv)=0thenfollowsthatb=c.Finally,λ=ccγ=c2.
Summarizingtheabove,forallv∈V,
2θ(v)=e1v4c+e2v2c+e3v3c+e4v1+v0c.
Hencewehaveθ(v)=vcforallvinthehyperplaneV:=e1c+e4,e2,e3,V0.
Buteverylineintersectsthishyperplane,thuscontainsafixedpoint.
V0=0:Inthiscasewemusthaveσ=id.FromnowonwewillassumeKis
commutativeandcharK=2.Sinceθpreservestheformq,wehaveλ=cσb=
bσc=λσ.Pickα∈K∗suchthatασ=−α.Thenvε:=e2+e3αisan
isotropicvectorcollineartobutdifferentfrome1,andθ(vα)=e2b+e3cαγ.
Byhypothesisthelinee1,vαcontainsafixedpoint.FromEquation3.4we
deducethatthisfixedpointmustbev,implyingαb=cαγ.Wedistinguish
cases:subowtγ=id:Fromαb=cαγfollowsb=c,thusasinthecaseV0=0oneverifies
thatθ(v)=vcforallvinthehyperplaneV:=e1c+e4,e2,e3,and
ws.folloclaimtheγ=id:Forallε∈FixK(σ)wehave(αε)σ=−αε,andsowemustinfacthave
αεb=c(αε)γ,i.e.,FixK(σ)⊆FixK(γ).Sinceγ=idonereadilyconcludes
thatγ=σ.Nowαb=cαγreducestob=−c.Butccγ=λ=cσb=−cγc,
acontradictionascharK=2.
Remark3.3.12.IfcharK=2andV0=0thentherearequadranglesthatsatisfy
theconditionsofProposition3.3.11yetstilladmitlineswithoutafixedpoint.
Moreover,itseemsquitelikelythattheassumptionthatKiscommutativeinthe
caseV0=0canbedropped.However,wedidn’ttryveryhardtoworkthisoutas
wehavetomakethisassumptioninotherplacesanyway.

54

3.3.Classicalquadrangles

3.3.4.Rθisalmostalwaysconnected
RecallthataPhanchamberisachamberwhichθmapstoanoppositeone.Wecall
apointresp.alinebadifitisnotfixedbutincidenttoafixedlineresp.point.We
callapointresp.alinegoodifitisneitherfixednorbad,equivalently,ifitisnot
incidenttoanyfixedelement.
Lemma3.3.13.LetRbeaMoufangquadrangle.Assumethatthepointandline
θneorctedersdifar(ep,grLe)ateriscthanonne4cte,dandwithinthatRaθtoPhaneverychambPhaner(p,chambL)erexists.(r,M)ThenRsatisfyingisctheon-
erties:opprlowingfol(1)MisoppositeL;
(2)p:=projL(r)isgood,hencep:=projLθ(p)isgood;
(3)r:=projM(p)isbad.
Proof.Let(s,K)beanarbitraryPhanchamber.Weprovethatitisconnectedto
(p,L)byagalleryofPhanchambers.
aIfgooLdpandoinKtonareK,equalisnotorconmeettainedinaingooLd.pSinceoint,wtheearelinedone.orderIfisnot,greaterthenthans,b4,eingby
Lemma3.3.6,siscontainedinatleasttwogoodlines.Sinceweareinaquadrangle,
itThefollochamwsbthaters(s,thereK)isandago(s,odK)lineareK(padjacenossiblyt,soKit=Ksuffices)tothroughconnectsnotthemeetinglatterLto.
(p,L)toestablishourclaim.
onKSincewhicthehpproointjectsordertoaisgoogreaterdpointhantp4on,bLy.IfLemmathepro3.3.7,jectiontherelineisisagogooodd,pweointarer
done.Otherwise,θitnowsufficestoconnectthePhanchambers(p,L)and(r,K)via
.RingalleryaDenotebyptheprojectionofθ(p)toL.Sincepisgood,p=p,andsorandp
goareodopplineosite.MthroughConsiderrthemeetingpencilsagoofordandlineofpthrough.Againp.byDenoteLemmabyr3.3.7,theprotherejectionisa

Figure3.1.:TheconstructionfromLemma3.3.13.

55

3.Flipsinrank1and2

ofptoM.Ifrisagoodpoint,wehaveconstructedaconnectionandaredone.If
itisbad,wecaninvokeourhypothesisandaredoneaswell.
Lemma3.3.14.LetRbeaclassicalquadrangleoverafieldK,letp,pbetwo
collineargoodpointssuchthatp,p,θ(p),θ(p)formathinsubquadrangle.Thenthe
hold:lowingfol(1)Thereareisotropicvectorse1,e2,e3,e4inVsuchthatp=e1,p=e2,
θ(p)=e4,θ(p)=e3.DenoteV0:=e1,e2,e3,e4⊥.Thereexistb,c,λ∈
K∗,D∈GL(V0),andγ∈Aut(K)oforderatmost2,suchthatforall
v1,v2,v3,v4∈K,v0∈V0,
γγγγγθ(e1v1+e2v2+e3v3+e4v4+v0)=e1λv4+e2cv3+e3bv2+e4v1+Dv0,
andλ=λγ=cbγ=bcγ=cσb=bσc,DDγ=λI.
(2)λ=λσ.
(3)ThefieldFixK(σ)∩FixK(γ)hasindexatmost4inK.
Proof.(1)ThefirstpartisaconsequenceofProposition3.3.8(2).Infact,letAbe
asin3.3.8(2),thenitimmediatelyfollowsthat
dA=c,wherea,b,c,d∈K∗.
baSinceθisonlydetermineduptoascalarfactor,wemaychoosea=1;from
AAγ=λI,wededuceλ=d=cbγ=bcγ=dγ,henceλ=λγ.
σ(2)Toseethatλ=λ,considertheisotropicvectorse1+e2,e3−e4,e1+e4and
e2−e4:
σ0=f(e1+e2,e3−e4)=f(θ(e1+e2),θ(e3−e4))=f(e3b+e4,−e1λ+e2c)=bc−λ
0=f(e1+e3,e2−e4)=f(θ(e1+e3),θ(e2−e4))=f(e2c+e4,−e1λ+e3b)=cσb−λ,
provingthatλ=cσb=(bσc)σ=λσ.
(3)WenowprovetheclaimthatFixK(σ)∩FixK(γ)hasindexatmost4inK.If
σ=id,thisistrivial.Else,pickx∈Kσandsetvx:=e1+e4x.Thenq(vx)=
x≡0(modK)σ,sovxisanisotropicvector.Thusalsoθ(vx)=e1λxγ+e4is
whenceisotropic,0=f(θ(vx),θ(vx))=f(e1λxγ+e4,e1λxγ+e4)=(λxγ)σ+λxγ.
Asλ=λσ,thisisequivalenttoxγσ=−xγ=xσγ,forallx∈Kσ.
DenoteF:=Fix(σ).IfcharK=2,thenKσ=F(itiseasytoseethatKσ⊆F;
butalsothatF∙Kσ=Kσ).Accordinglyγinducesaninvolutoryautomorphism

56

3.3.Classicalquadrangles

ofF,andsoFixK(σ)∩FixK(γ)hasatmostindex2inF,ergoatmostindex4
.KinIfcharK=2,pickα∈K∗suchthatασ=−α.WeclaimthatKσ=αF:If
x∈Kσ,thenx=tσ−tforsomet∈K,hencexσ=−x.Ontheotherhand,if
x∈αF,thenx=tσ−tfort=−2x.
WealreadyknowthatσandγcommuteonKσ=αF.Ify∈F,theny=
(α−1)(αy).Sinceα−1,αy∈αF,wehaveyγσ=yσγaswell.Henceσandγ
commuteonFaswell,andtheclaimfollows.
TheprecedingtwolemmasfinallyenableustoproveconnectednessofRθunder
theassumptionthataPhanchamberexists.
Proposition3.3.15.LetRbeaclassicalquadrangleoverafieldK,letθbean
involutoryautomorphismofR.IfθadmitsaPhanchamber,and|K|>9,|K|=16,
thenRθisconnected.
Proof.ByLemma3.3.13,itsufficestofixaPhanchamber(p,L),andthenprove
thatitisconnectedtoeveryPhanchamber(r,M)satisfying
(1)MisoppositeL;
(2)p:=projL(r)isgood,hencep:=projLθ(p)isgood;
(3)r:=projM(p)isbad.
Sincepandp=projLθ(p)aregoodandcollinear,Lemma3.3.14yieldsadescrip-
tionofθandfusingaconvenientbasisoftheunderlyingvectorspaceV.
WeconstructthedesiredconnectionbyshowingthatthereisagoodpointonM
whichprojectstoagoodpointonLviaagoodprojectionline.Forthis,weneedto
characterizethreesubsetsofthepointrowofM:Thebadpoints;thepointswhich
projecttoabadpointonL;andthoseforwhichtheprojectionlinetoLisbad.
Theprojectionsofp=e1resp.p=e2toMarer=mresp.r=m.
Furthermore,pisnotcollineartor,andpisnotcollineartor.Forµ∈Klet
xµ:=mµ+m.Thenxµ∈M,andallpointsofMexceptforrareobtainedin
thisway.TheprojectionofxµtoLisyµ,whereyµ:=e1µσ−e2.
WenowstudyindetailthethreesubsetsofMmentionedearlier(actually,for(2)
and(3)weleftoutoneelement,butthatisirrelevantforourpurposes):
(1)ThesetA:={µ∈K|xµisabadpoint},i.e.,thesetofallµforwhichxµis
collineartoθ(xµ),correspondstothesolutionsofthefollowingequation:
0=f(xµ,θ(xµ))=f(mµ+m,θ(mµ+m))
=f(mµ,θ(mµ))+f(mµ,θ(m))+f(m,θ(mµ))+f(m,θ(m))
=µσf(m,θ(m))µγ+µσf(m,θ(m))+f(m,θ(m))µγ+f(m,θ(m)).

57

3.Flipsinrank1and2

Sincer=m=x0isbad,f(m,θ(m))=0.So
A={µ∈K|µσγµ∙f(m,θ(m))+µσγ∙f(m,θ(m))+µ∙f(m,θ(m))=0}.(3.5)
NotethatAisapropersubsetofK,asr=misagoodpoint,implying
f(m,θ(m))=0,hencethedefiningequationofAdoesnotvanisheverywhere.
(2)ThesetB:={µ∈K|yµisabadpoint},i.e.,thesetofallµforwhichyµ
iscollineartoθ(yµ),correspondstothesolutionsofthefollowingequation:
0=f(yµ,θ(yµ))
=f(e1µσ−e2,−e3b+e4µσγ)
=µµσγ+b
=⇒B={µ∈K|µµσγ+b=0}.(3.6)
Notethatp=e2=y0isagoodpoint,andindeed,clearly0∈/B.
(3)ThesetC:={µ∈K|xµ,yµisnotagoodline},canbecharacterizedas
follows:Thelinexµ,yµisnotgoodifitcontainsafixedpoint;equivalently,
apointwhichiscollineartobothθ(xµ)andθ(yµ).Thusthelinexµ,yµisnot
goodifandonlyifthereexists(α,β)∈K2\{0}suchthat
f(xµα+yµβ,θ(xµ))=0=f(xµα+yµβ,θ(yµ))
αf(xµ,θ(xµ))f(yµ,θ(xµ))
⇐⇒Zµ(β)=(00),whereZµ:=f(xµ,θ(yµ))f(yµ,θ(yµ)).
Butthisisequivalenttodet(Zµ)=0.Hence
C:={µ∈K|f(xµ,θ(xµ))∙f(yµ,θ(yµ))=f(yµ,θ(xµ))∙f(xµ,θ(yµ))}.(3.7)
Wenowarguethatdet(Z0)=0andhence0∈/C:Weknowthatr=m=
x0isbad,hencef(x0,θ(x0))=0.Butf(y0,θ(y0))=0sincep=e2=y0
isgood.Moreoverf(y0,θ(x0))=0,forelsey0wouldbecollineartoand
differentfrombothx0andθ(x0),whicharedistinctbutcollinearpoints,
consequentlyy0,x0,θ(x0)wouldformatriangle,whichisimpossible.
LetF:=FixK(σ)∩FixK(γ).ByLemma3.3.14,FisasubfieldofKofindex
atmost4.OverF,thedefiningequationsforthesetsA,BandCbecomenonzero
polynomialequationsinµofdegree2,2and4,respectively(asthetermsf(xµ,θ(xµ)),
f(xµ,θ(yµ))etc.becomequadraticpolynomialsoverF).Inparticular|(A∪B∪C)∩
F|≤2+2+4=8.Thusif|F|>8,thereexistsµ∈Fsuchthatnoneoftheequations
hold.Butthenxµ,yµandxµ,yµallaregood.Forthisreason,(r,M)and(p,L)
areconnectedif|K|>84=4096,inparticularoverallinfinitefields.
IfKisafinitefield,itiswell-knownthatAut(K)iscyclicandgeneratedbythe
Frobeniusautomorphismx→xp,wherep=charK.HenceKadmitsa(necessarily
unique)involutoryautomorphismsifandonlyiftheorderofKisasquare.

58

3.3.Classicalquadrangles

If|2K|isnotasquare,qF=Kandwegetaconnectionif|K|>8.If|K|isasquare,
saandyq3.6,arethenpx→olynomialxistheequationsuniqueinµofautomorphismdegreeatofmostorderq2.+1,HenceandEquationsEquation3.73.5
isapolynomialequationofdegreeatmost2q+2,2implying|A∪B∪C|≤4q+4.
andThereforethus|wK|e>can16.constructaconnectionif|K|=q>4q+4,equivalentlyifq>4
Finally,weprovethatRθisalwaysconnectedifnoPhanchamberexists.
Proposition3.3.16.LetRbeaclassicalquadrangleoverafieldK,charK=2,let
θRθbeisanconnecteinvolutoryd.automorphismofR.IfθdoesnotadmitaPhanchamber,then
Proof.SincenoPhanchamberexists,wecanuseProposition3.3.11toconcludethe
following:Eitherapointpexistswhichisoppositetoθ(p).Thenalllinesthroughp
pmisustconcollineartainatofixeditspimage.oint(elseThenwepwisouldinfactgetaconPhantainedchaminbaer).lineOrfixedelsebyevθ:eryIfppoinist
notarbitraryfixed,pointhentpp,θ(pcollinear)istothebut(unique)differentfixedfromlinep=θ(throughp),thenp.pIfpisisalsofixed,collineartaketoan
θ(p);sincenotrianglesexist,weconcludethatθthenfixesalllinesthroughp.
Souptodualitywemayassumethatallpointsarecontainedinafixedline.If
θfixesallpoints,itistheidentityandwearedone.Else,pickanon-fixedpointp.
one.ThenWaneylineconcludeLthatthroughRθpconsistsdifferenoftallfromchamp,θb(pers)ofisthismappkind:edbAyθtonon-fixedanopppoinositetp
andalineLwhichismappedtoanoppositelineθ(L).
RθGivasenfollotwows:sucIfhLc1hamandbLers2(conp1,Ltain1)aan(pcommon2,L2),pwoinet,consthentructitisanecessarilyconnectionawithinnon-
fixedpointandwehavethedesiredconnection.Iftheyareopposite,twocasesare
possible:First,thereisanon-fixedprojectionlinefromapointonL1toapoint
onL2,yieldingaconnection(theintersectionpointsmustbenon-fixed,andthe
protriangles).jectionOrline,bsecondlyeing,allnon-fixed,projectioncannotlinesconaretainafixed.fixedInpointhatt,ascase,therepickcanablineeLno
2thicthroughkp2quadrangle).whichisThendifferentheretfrommustL2beandpnon-fixed2,θ(p2)pro(itjectionexistslinesbbecauseetwweeneLare1inanda
L2,whencewehavereducedtothefirstcaseandaredone.

59

3.

60

Flips

in

rank

1

and

2

CHAPTEROURF

STRUCTUREOFFLIP-FLOPSYSTEMS

Throughoutthischapter,C=(C+,C−,δ∗)isatwinbuildingoftype(W,S),andθis
.ofquasi-flipaC

4.1.Flip-flopsystems
InthischapterwestudycertainchambersubsystemsofCwhichareassociatedto
.θquasi-flipbuildingtheOntheonehand,westudyminimalPhanresidues(recallthataPhanresidue
isaresiduewhichismappedbyθtoanoppositeone).Insomesense,thestudy
ofminimalPhanresiduesislocal.Forexample,apriori,wecannotrelatetwo
differentminimalPhanresidues.Moreover,onemightconsistofasinglechamber
whileanothercouldbeamuchlargerresidue,possiblyevennon-spherical.
Despitethis,wewillshowthatundersuitableassumptions,allminimalPhan
residueshaveidenticaltype.Fromthepointofviewofgroups,thisisequivalentto
allminimalθ-splitparabolicsubgroupshavingequaltype.1Thisisknowntobetrue
foralgebraicinvolutionsofalgebraicgroups,seee.g.[HW93].
Ontheotherhand,westudytheso-calledflip-flopsystemconsistingofallchambers
whicharemappedasfarawayaspossible,globally.Tomakethisprecise,consider
definitions:wingfollotheDefinition4.1.1.Letθbeaquasi-flipofatwinbuildingC,letRbeanyresidueof
C+(inparticular,RmightequalC+).Theminimalnumericalθ-codistanceofR
isthevalueminc∈Rlθ(c)=minc∈Rl(c,θ(c)).
Withtheabove,thesetofchamberswhicharemapped“asfarawayaspossible”
byaquasi-flipcanbedescribedasfollows.
1RecallthataparabolicsubgroupPisθ-splitifP∩θ(P)isamaximalLevifactorinbothPand
.)P(θ

61

4.Structureofflip-flopsystems

C+.DefinitionTheinduced4.1.2.Letflip-flopθbeasystequasi-flipmRθofonaRtwinassociatedbuildingtoC,θletisRthebea(sub)cresiduehambofer
systemRθ:={c∈R|lθ(c)=d∈minRlθ(d)}
withflip-floptheequivsystemalenceassociatedrelationstoθisinheritedthe(sub)cfromCham+.berInsystemparticular,Cθ:=Cforθ.R=C+the
+Forthisgloballydefinedchambersubsystem,wecannowforexampleaskwhether
itisquestion.connected.MoreovFer,orawewideshowclassthatofintwinmanybuildingcases,s,thewegivflip-flopeapositivsystemeanswcoincidesertowiththis
theminimalunionofPhanallresiduesminimalwePhanalreadyresidues.mentionedThisabthenove.yieldsthehomogeneityresulton
mericalRemarkθ-co4.1.3.distanceIfθisis1aW,asstrongtherequasi-flip,existPhanthencbhamybeLemmars.Moreo2.4.2,ver,theminimalminimalnPhanu-
residuesalwaysarePhanchambers.Sointhiscase,homogeneityindeedholds.
Example4.1.4.Thefollowingexampleoriginallycomesfrom[BS04].
fieldLetVbeautomorphisman(n+σ1)oforder-dimensional2.vConsiderectorspacthee(nbuilding≥2)CovofertaypefiniteAfieldassoFciatedwithtoa
nVof,noni.e.,trivialthepropropjectiveresubspacesspaceP(ofV)V..Here,FixathebasischamofbV.ersThearegroupmaximalSL(Vflags)=SLnconsisting+1(F)
actsstronglytransitivelyonthebuildingC,whereB+andB−,thesubgroupsof
upperresp.lowertriangularmatrices,stabilizeoppositechambers.
1−σt(asTheitinσ-ttercwistedhangesBChev+andalleyB−in).vTheolutionθinduced:x→prop(erx)buildingisapropquasi-fliperBNsendsv-quasi-flipector
subspacestotheirorthogonalcomplementwithrespecttothestandardσ-sesquilinear
formfonV(“standard”regardingourfixedbasis).
mappTheedtoflip-flopoppositesystemflags.thenTheseconsistsmaximalofallcflagshamarebersprecisely(i.e.,thomaximalsewhereflags)allwhicinvholvareed
vectorsubspacesarenon-degeneratewithrespecttof.Itisnothardtoseethatany
flagconsistingofnon-degeneratesubspacescanbeextendedtoachamber(amaximal
flagtheflip-flopconsistingofsystemnisproperresiduallynon-trivialconnectedifsubspaces).|F|>I4n.fact,by[BS04,Corollary2.4],

Onceweunderstandhomogeneityandconnectedness,weturntothequestion
whetherthechambersystemCθisresiduallyconnected(cf.Section1.4).Weestablish
thisforthespecialcasethattheflipisK-homogeneous(meaningallminimalPhan
chamresiduesberhavsystemetypCeθK)(seeand|KDefinition|≤2.In4.5.1)isgeneral,wresiduallyeprovethatconnected.theFroso-calledmthis,Kresidual-residue
connectednessofCKθwouldfollowifonecouldsolveaprobleminCoxetersystems
details).for4.5Section(see

62

4.1.systemsopFlip-fl

TheimportanceofthisisthatwhenCθisresiduallyconnected,onecanconstruct
aso-calledsyntheticorincidencegeometryfromit(inthesenseofBuekenhoutand
Tits,seee.g.[BC]),theflip-flopgeometryGθ:=G(Cθ)associatedtoθ.Inall
examplesknowntous,Gθisinfactageometry.
Thefollowingpropertyisthecornerstoneofourapproachtoprovingalltheresults
hintedatabove:
Definition4.1.5.LetRbearesidue.WesaythatdirectdescentintoRθis
pθossibleifforanychambercθinRthereexistsagalleryinRfromctoachamberin
Rwiththepropertythatl(asdefinedinSection2.1)isstrictlydecreasingalong
.gallerytheRemark4.1.6.Ifθisastrongquasi-flip,thenbyLemma2.4.2directdescentis
possibleforallresidues.Infact,quasi-flipswhichallowdirectdescentforallresidues
maybethoughtofasgeneralizingstrongquasi-flips.
Indeedin[DM07],severaloftheresultswepresentherehavebeenelegantlyproven
forstrongflipsusingfiltrationsofbuildings.Infactthelocal-to-globalconnectivity
andhomogeneityresultsinthischaptercanbeconsideredasgeneralizationsofcor-
respondingresultsforstrongflipsinloc.cit.;butinaddition,weperformarank2
analysiswhichcouldindependentlybecombined(forstrongflips)withtheDevillers-
filtration.MühlherrThefollowingholds(notethatwedonotrequirethetwinbuildingtobeMoufang):
Theorem4.1.7(jointworkwithGramlichandMühlherr).Letθbeaquasi-flipofa
twinbuildingCsuchthatforallrank2residuesR,directdescentintoRθispossible
andRθisconnected.
Thentheflip-flopsystemCθisconnectedandequalstheunionofallminimal
Phanresidues.TheminimalPhanresiduesallhaveidenticalsphericaltypeK,or
equivalently,δθtakesontheconstantvaluewKonallchambersofCθ,wherewKis
thelongestelementofthesphericalCoxetersystem(WK,K).Moreover,thechamber
systemCKθofK-residuesofCθisconnectedandresiduallyconnected.
Proof.CombineProposition4.4.4and4.5.3.
InSection4.6weinvestigatecloserforwhichrank2Moufangbuildingsthecondi-
tionsofTheorem4.1.7aresatisfied.Thisculminatesinthefollowingtheorem,which
weproveinSection4.7:
Theorem4.1.8(jointworkwithGramlichandMühlherr).Letθbeaquasi-flipof
anRGD-system(G,{Uα}α∈Φ,T)oftype(W,S),whereallrootgroupsUαareuniquely
2-divisible.Assumethediagramissimplylaced;orassumethattheRGD-systemis
2-spherical,F-locallysplit,|F|>4,andnoG2residuesoccur.
Thenforallrank2residuesR,directdescentintoRθispossibleandRθiscon-
d.ctene

63

4.Structureofflip-flopsystems

Theorem4.1.8shouldinfactextendtomost2-sphericalbuildings(withtheex-
ceptionofsomesmallrank2cases).ThisissubjectofongoingresearchbyHendrik
VanMaldeghemandtheauthor[HVM].Weconjecturethefollowing:
Conjecture4.1.9.Letθbeaquasi-flipofanRGD-system(G,{Uα}α∈Φ,T)oftype
(W,S),whereallrootgroupsUαareuniquely2-divisible.Assumethediagramis
2-spherical,andallrank2residuesarenotincludedinafinitelistofexceptions.
Thenforallrank2residuesR,directdescentintoRθispossibleandRθiscon-
d.cteneCombiningTheorems4.1.7and4.1.8weconclude.
Theorem4.1.10.Letθbeaquasi-flipofanRGD-system(G,{Uα}α∈Φ,T)oftype
(W,S),whereallrootgroupsUαareuniquely2-divisible.Assumethediagramis
simplylaced;orassumethattheRGD-systemis2-spherical,F-locallysplit,|F|>4,
andnoG2residuesoccur.
Thentheflip-flopsystemCθisconnectedandequalstheunionofallminimalPhan
residues,whichinturnallhaveidenticalsphericaltypeK.Thechambersystemof
K-residuesofCθisconnectedandresiduallyconnected.
Remark4.1.11.Thispartiallyanswersthequestionposedin[BGHS03]regarding
whethertheflip-flopsystemisgeometricingeneral;residualconnectednessimplies
this.

4.2.Outlineoftheproof
InSection4.3weprovethefollowingfacts,withoutanyassumptionsonthetwin
buildingorthequasi-flip:AnyminimalPhanresidueRisspherical,andifRisof
typeJ,theθ-codistancemustbeconstantandequaltothelongestelementofWJ.
InSection4.4weassumethatforanyrank2residueR,theinducedflip-flopsystem
Rθisconnected,andthatdirectdescentintoRθispossible.Undertheseassumptions,
CθishomogeneousandinheritsconnectednessfromC+,asdefinedbelow:
Definition4.2.1.Aquasi-flipθiscalledhomogeneousorK-homogeneousifall
minimalPhanresidueshaveidenticaltypeK.
Definition4.2.2.LetC,CbechambersystemssuchthatC⊆Candtheequiv-
alencerelationsonCareobtainedbyrestrictingthoseonC.WesayCinherits
connectednessfromCifanytwochambersc,dinCareconnectedbyaJ-gallery
inCifandonlyiftheyareconnectedbyaJ-galleryinC.
AssumewearegiventwodistinctminimalPhanresiduesRandRoftypesI
andI,resp.,andpickchamberscandcfromeach.Chooseanyminimalgallery
connectingthetwoanddenoteitstypebyJ.WedeformthisJ-galleryviaaseries
oflocaltransformations(insiderank2residues)toanewJ-galleryγonwhichthe

64

residuesPhanMinimal4.3.

ncoumericaldistanceθis-coconstandistancet.isHenceconstanRandt.RThismustthenhaveimpliesbeenbyofLeequalmmatyp4.4.1eI=thatI.theSoCθ-θ
isI-homogeneousandinheritsconnectednessfromC+asclaimed.
θInnectednessSectionfrom4.5Cwe,prothenvethethecfollohamwbering:IfsystemCisCθKofK-homogeneous-residuesofandCθisinheritsresiduallycon-
+connected,whichwillcompletetheproofofTheoremK4.1.7.
InSection4.6weturntostudyingwhathappensinrank2residues,withthegoal
ofdeterminingexplicitcriteriaontherank2residues,whichimplydirectdescentinto
andconnectednessoftheinducedflip-flopsystems.Here,weassumetheMoufang
arepropinertyfactontheMoufangbuilding,polygons.whichForimpliesexample,thatfortheA2rankresidues2residuesnotinwceneedharacteristictostudy2,
hold.ertiespropdesiredtheFinally,inSection4.7theproofofTheorem4.1.8andtheothermainresultsof
thischapterarepresented.

iduesresPhanMinimal4.3.RecallthataPhanresidueisaresiduewhichθmapstoanoppositeresidue.Inthis
sectionwecharacterizePhanresidueswhichareminimalwithrespecttoinclusion.
Lemma4.3.1(Lemma5.140(1)in[AB08]).Forε∈{+,−}letx∈Cεandy,z∈
Then.ε−Cδ∗(x,z)≤δ∗(x,y)∙δ−ε(y,z)
der.orBruhattheinInLemma2.3.1wesawthatifwistheθ-codistanceofachamber,thenwisa
θ-twistedinvolution,soθ(w)=w−1.Wewillmakefrequentuseofthisfactinwhat
w.elobwsfolloLemma4.3.2.LetRbeaPhanresidueoftypeI.Thentheθ-codistanceonRhas
.WinimageIProof.Takeanychamberc∈Randletw=δθ(c)denoteitsθ-codistance.Since
RisaPhanresidue,thereexistsd∈Roppositeθ(c).ApplyingLemma4.3.1with
x=θ(c),y=d,z=cyields
w−1=θ(w)=δ∗(θ(c),c)≤δ∗(θ(c),d)∙δ−ε(d,c)=1W∙δ−ε(c,d)−1.
Sincec,d∈R,wegetw∈WIandtheclaimfollows.
Lemma4.3.3.LetR1andR2bePhanresiduesoftypeI1andI2withnonempty
intersection.ThenR1∩R2isalsoaPhanresidue.

65

4.Structureofflip-flopsystems

Proof.Pickanychamberc∈R1∩R2.Itsufficestofindachamberd∈R1∩R2
suchthatδ∗(c,θ(d))=1W.ByLemma4.3.2andsincec∈R1∩R2,wehave
w:=δθ(c)∈WI1∩WI2=WI1∩I2.Walkingfromcalonganygalleryoftypew,we
arriveatachamberdwithδ+(c,d)=w.ApplyingLemma4.3.1withx=c,y=θ(c)
andz=θ(d),wededuce
δ∗(c,θ(d))≤δ∗(c,θ(c))∙δ−(θ(c),θ(d))=w∙θ(δ+(c,d))=w∙θ(w)=1W,
henceδ∗(c,θ(d))=1W.AccordinglyR1∩R2isaPhanresidue.
Lemma4.3.4.Minimal(byinclusion)Phanresiduesarespherical.Inparticular,
sphericalPhanresiduesexist.
Proof.LetRbeaPhanresidueoftypeJ.TakeanychambercinR,anddenote
itsθ-codistancebyw.ByLemma2.3.4,thereexistasphericalsubsetIofS,an
elementw∈Wwithw≤wintheBruhatorder,andachamberc∈C+suchthat
δ+(c,c)=wandδθ(c)=wI.ByLemma4.3.2,w∈WJ.Asw<wintheBruhat
order,itiscontainedinWJ.Hencec∈R.
TheI-residueRI(c)aroundcisspherical.Moreover,itisaPhan-residue:Pick
achamberd∈RI(c)suchthatδε(c,d)=wI,henceδ−ε(θ(c),θ(d))=θ(wI)=
wI−1=wI(thelatterequalityholdsbecausewIisaθ-codistance,henceaθ-twisted
involution,butalsoaregularinvolution).ApplyingLemma4.3.1withx=c,y=
θ(c),z=θ(d),yieldsδ∗(c,θ(d))=1W.HenceeveryPhanresiduecontainsspherical
Phanresidues,andtheclaimfollows.
Lemma4.3.5.LetRbeaminimalPhanresidueoftypeI.ThenIissphericaland
theθ-codistanceonRisconstantandequaltowI,thelongestelementofWI.
Proof.ThatIissphericalfollowsfromLemma4.3.4.Assumethatthereexistsa
chamberc∈Rsuchthattheθ-codistancewofcisdifferentfromwI.
ByLemma2.3.4,thereexistasphericalsubsetJofS,anelementw∈W,and
achamberc∈C+suchthatδ+(c,c)=wandδθ(c)=wJ.Moreover,w≤wand
wJ≤wintheBruhatorder.YetbyLemma4.3.2,w∈WI,andbyassumption
w=wI,thuswJ≤w<wIandsoJI.ThentheJ-residueRJ(c)aroundc
wouldbeaPhan-residuecontainedinR,butstrictlysmallerthanit,contradicting
theminimalityofR.
Asafirstimmediateapplication,wehighlighthowK-homogeneityinfluencesthe
system.flip-floptheofstructureLemma4.3.6.IfθisaK-homogeneousquasi-flip,thenforallchambersc,wK≤
δθ(c)intheBruhatorder,andtheflip-flopsystemCθequalstheunionofallminimal
esidues.rPhanProof.Letcbeanarbitrarychamber,denoteitsθ-codistancebyw.
ByLemma2.3.4,thereexistasphericalsubsetJofS,anelementw∈W,anda
chamberc∈C+suchthatδ+(c,c)=wandδθ(c)=wJ.Moreover,wandwJare
lessorequalwintheBruhatorder.

66

4.4.Homogeneityandinheritedconnectedness

NosphericalwassumePhanwKresidue≤wJ,whichthereforconetainsK⊆noJK.It-residue.followsButthatbytheKresidue-homogeneitRJ(cy,)isanya
wKPhan≤wJresidue≤w=conδθ(tainsc).aPhanresidueoftypeK,whichisacontradiction.Hence
Inparticular,ifc∈Cθ,thenwK≤δθ(c);butsincethenumericalθ-codistanceof
θcPhanisgloballyresidueoftminimal,ypeKw.emMoreoustviner,factbyhaveLemmaδ(c)4.3.5=wwKe.havTheneRKR(Kc()c)⊂isCaθ,sominimalCθis
theunionofallminimalPhanresidues.

4.4.Homogeneityandinheritedconnectedness
Wenowestablishthattheflip-flopsystemishomogeneousandinheritsconnectedness
ifforallrank2residuesR,directdescentintoRθispossibleandRθisconnected.
First,weprovealittlelemmawhichshowsthatontheconnectedcomponentsofthe
flip-flopsystemtheθ-codistanceisconstant.
Lemma4.4.1.Iftwoadjacentchambershaveequalnumericalθ-codistance,then
theyhaveequalθ-codistance.
Proof.Considertwos-adjacentchamberscandcwithequalnumericalθ-codistance
andsetv:=δ∗(c),v:=δ∗(c).Wehavel(v)=l(v)byassumption.Butthenv=v
(andwearedone),orv=svθ(s).ButbyLemma1.3.2,l(svθ(s))=l(v)implies
sv=vθ(s)andhencev=vafterall.
Theactualheartofourproofisthefollowingusefullemma.
Lemma4.4.2.Letθbeaquasi-flipofaθtwinbuildingCsuchthatforallrank2
residuesR,directdescentispossibleandRisconnected.
Let(c0,c1,c2)beagallerysuchthatthenumericalθ-codistanceofc1isatleastas
bigasthatofc2andexceedsthatofc0.Thenthereexistsagalleryγfromc0toc2
suchthatthenumericalθ-codistanceofallchambersinγ\{c2}islowerthanthatof
c1(seeFigure4.1).

Figure4.1.:Apeakandashortplateauwithbypassesintheθ-codistanceofagallery.
Highernumericalθ-codistanceisreflectedbychambersbeingdepicted
ards.wupfarther

67

4.Structureofflip-flopsystems

Proof.Clearlythereisarank2residueRwhichcompletelycontains(c0,c1,c2).
SincebyhypothesisdirectdescentintoRθispossibleinR,thereisagalleryγ0
fromc0toachamberc0inRθsuchthatlθisstrictlydecreasingalongthisgallery.
Inparticular,thenumericalθ-codistanceofanychamberinthisgalleryisatmost
equaltothatofc0andhencestrictlylessthanthatofc1.Likewisewefindagallery
γ2fromc2toachamberc2inRθonwhichlθisstrictlydecreasing.Henceforall
chambersinthisgallerydifferentfromc2,thenumericalθ-codistanceisstrictlyless
thanthatofc2andhencec1.
Finally,Rθisconnectedbyhypothesis,thereforethereexistsagalleryγ1inRθ
connectingc0andc2.Weconcludethatγ=γ0γ1γ2−1isagallerywiththedesired
erties.propThekeyideaisnowtorepeatedlyinvoketheprecedinglemma.
Proposition4.4.3.Letθbeaquasi-flipofatwinbuildingCsuchthatforallrank
2residuesRdirectdescentispossibleandRθisconnected.ThenforanyresidueQ
ofC+,directdescentispossibleandQθinheritsconnectednessfromQ.Inparticular,
Qθisconnected.
Proof.Chooseanyc∈Qandd∈Qθ.Pickaminimalgalleryγ=(c0,c1,...,cn)
betweencandd,soc0=candcn=d.ByminimalityofγandconvexityofQwe
haveγ⊆Q.
Ourgoalistotransformγviaaseriesoflocaltransformationsintoagalleryγsuch
thatthenumericalθ-codistanceofcisatleastasbigasthatofanyotherchamber
inγ,andsuchthatifγisaJ-gallery,thensoisγ.
Denotebymthemaximalnumericalθ-codistanceamongallchambersinγ,and
denotebyXγthesetofallchambersinγwithnumericalθ-codistancem.Assumec=
c0∈/Xγ.LetcibethechamberfromXγwhichisclosesttoc0alongγ,andconsider
thesubgallery(ci−1,ci,ci+1).Byourchoice,lθ(ci−1)<lθ(ci),andwecanapply
Lemma4.4.2toobtainagalleryγfromci−1toci+1bypassingciandcontainingno
chambersinXγexceptforpossiblyci+1.Nowsubstitutethesubgallery(ci−1,ci,ci+1)
inγbyγtoproduceanewgallerywithoneelementlessatnumericalθ-codistance
m.Werepeatthisprocess|Xγ|times,untilwearriveatagalleryγbetweenc0and
cnwithmaximalnumericalθ-codistancestrictlylessthanm.
Takenowγasournewgalleryγ.Repeatingtheabovefinitelymanytimes
(boundedbytheinitialvalueofm),wearriveatagalleryγ⊂Qwheretheset
Xγcontainsc0=c.
Hencewemayassumefromθnowonthatc∈Xγ,soθallchambersinγhave
numericalθθ-coθdistanceatmostl(c).Ifinadditionc∈Q,thenthisimpliesthat
γ⊂QandQinheritsconnectednessfromQ(asourconstructionabovetransforms
J-galleriesintoJ-galleries).
Toprovethatdirectdescentispossible,weproceedbyinductiononn:=lθ(c)−
lθ(d).Ifn=0,thenc∈QθanddirectdescentfromcintoQθistriviallypossible.

68

4.5.Residualconnectedness

Assumenowthatn>0,andthatdirectdescentintoQθispossibleforallchambers
withnumericalθ-codistancelessthanlθ(c).Sincelθ(c)>lθ(d),wehaved=cn∈/Xγ.
Denotebyithelowestindexsuchthatci∈Xγbutci+1∈/Xγ.
Ifi>0,weapplyLemma4.4.2,thistimecomingfromtheright,to(ci+1,ci,ci−1)
(clearlyci+1haslowernumericalθ-codistancethanc0).Thisyieldsagalleryγfrom
ci+1toci−1whichbypassesciandallchambersinγexceptforci−1havelowernumer-
icalθ-codistancethanc0.Defineanewgalleryγbysubstitutingγfor(ci+1,ci,ci−1)
inγBy.Inrepγ,eatingclearlythisci−1pronocesswihastimes,thewsameearrivpropeatertyaascgalleryihadinwhereγ.c=chasbigger
0nintoQumericalθispθ-coossible,distancethuswthaneccan1.Byalsoourdirectlyinductiondescendhypfromothesis,c0intodirectQθ.descentfromc1
Thefollowingisanimmediateconsequence.
2rPropesiduesositionRdire4.4.4.ctdescLetentθbisepaossiblequasi-flipandofRθaistwinconnebuildingcted.CThensuchθisthathomoforalgenelrousank
andCθinheritsconnectednessfromC+.
Proof.ApplyingPropθosition4.4.3toQ=C+,wefindthatθdirectdescentintoCθis
possibleandthatCinheritsconnectednessfromC+.HenceCisconnected,andso
byLemma4.4.1theθ-codistanceonCθisconstantandequalssomeelementw∈W.
ByLetLemmaRbean2.3.4,therearbitraryexistsminimalK⊆SPhansuchthatresidueKofistypesphericalI.ByandwLemma=wK.4.3.5,Iis
sphericalandtheθ-codistanceonRisconstantandequaltowI.Butfromany
chambercinRwecandirectlydescendintoCθ.Yettheonlywayonecouldshorten
theelementwIiswithsomes∈I,henceweactuallystayinsideR,wherethe
θ-codistanceisconstant.AltogetherthisprovesthatR⊂Cθ,I=Kandθis
-homogeneous.K

connectednessidualRes4.5.Iamnotawareofagoodreferenceforthefollowingdefinition,buttheobjects
describedinitarecertainlynotnew.Seeforexample[BC,Theorem14.6.3]fora
definition.similarweDefinitiondefinethe4.5.1.KLet-residueCbceahamcbhamerbersystemsystemCoKverovIer,Ilet:=KbIe\aKassubsetfolloofIws:.TheThen
cofhamCbareersiare-adjacentheKtifand-residuesonlyRKif(bc)othinCare,andconfortainedi∈inI,thetwocsameham(Kb∪ers{iR})K(c),-residue.RK(d)
KRemark4.5.2.ThiscorrespondstogoingfromthebuildingG/Btothecoset
case,geometryPisinG/Pfact,awhereminimalPisθa-splitparabparabolicolicsubgroupsubgroup.oftypeKGeometricallycontaining,weBmo.veInfromour
apartialcompleteflagvflagarietvyariety(where(sayw,eallomitpropsomeertnonypestrivialofsubspaces,subspacesasofavectdeterminedorspace)byKto).a

69

4.Structureofflip-flopsystems

Proposition4.5.3.LetθbeaK-homogeneousquasi-flipofaMoufangtwinbuilding
Coftype(W,S)θsuchthatCθinheritsconnectednessfromC+.ThentheK-residue
chambersystemCKisresiduallyconnected.
Proof.LetI=S\KdenotethetypesetofCKθ.ToestablishthatCKθisresidually
connected,consideranarbitraryfinitefamilyofresidues(Rj)j∈JinCKθ,whereJ⊆
IandeachRjisoftypeI\{j},andsuchthatallRjhavepairwisenonempty
intersections.WehavetoprovethattheintersectionofallRjisnonempty.
ForeachjinJ,definethecompletionRjofRjinC+astheunique(S\{j})-residue
ofC+whichcontainsRj(viewedasasubsetofC+).ThesecompletionsarePhan
residues(i.e.,Rjisoppositeθ(Rj)),fortheycontainK-residuesofchambersinCθ.
SinceC+isresiduallyconnected,RJ:=∩j∈JRjisanonempty(S\J)-residueof
C+.ByLemma4.3.3,itisagainaPhanresidue,hencecontainsaminimalPhan
residueoftypeK.UsingLemma4.3.6,weconcludethatitintersectsCθnontrivially.
SetRJ:=θRJ∩Cθ.Letc∈RJandj∈Jbearbitrary.Pickachamberdin
Rj⊂Rj⊂C.Byθvirtueoftheirdefinition,canddareconnectedbyan(I\{j})-
galleryinC+.AsCinheritsconnectednessfromC+,theyarealsoconnectedbyan
(I\{j})-galleryinCθ,andweseethatc∈Rj.Sincecandjwerearbitrary,itfollows
thatRJ⊂∩j∈JRj.ButRJisnonempty,hencetheclaimfollows.
IfK=∅,thenProposition4.5.3statesthatCθitselfisresiduallyconnected.If
K=∅,itisingeneralunknownwhetherthisisthecase.However,forthespecial
casethat|K|≤2andthatwehavedirectdescentinallresidues,thisistruebya
simpleargumentinCoxetergroups,asthefollowingshows:
Proposition4.5.4.Letθbeaquasi-flipofatwinbuildingCoftype(W,S)such
thatforallrank2residuesR,directdescentispossibleandRθisconnected.Ifθis
K-homogeneouswith|K|≤2,thenCθisresiduallyconnected.
Proof.Fori∈{1,2,3},letRibearesidueoftypeJiinCθ,suchthatforeach
j∈{1,2,3}wehaveRi∩Rj=∅.ToshowresidualconnectednessofCθ,byLemma
1.4.11itsufficestoshowthatR123:=R1∩R2∩R3isnonemptyandconnected.
DenotebyRitheconvexhullofRiinC+.Byresidualconnectednessofthebuilding,
wehaveR123:=R1∩R2∩R3=∅,sowecanpickachambercinR123.Applying
Proposition4.4.3toR1∩R2wecandirectlydescendfromcintoCθviaa(J1∩J2)-
gallery.Bysymmetrywecandolikewiseviaa(J2∩J3)-galleryora(J1∩J3)-gallery.
Moreprecisely,anddenotingtheθ-codistanceofcbyw,thismeansthatthereare
wordswij∈WJi∩Jjsuchthatforeachthereisadirectlydescendinggalleryoftype
wijfromcwithθ-codistancewtoachamberwithθ-codistancewK.Hencethereare
subwordswij≤wijsuchthatw=wijwKθ(wij)andl(w)=l(wij)+l(wK)+l(θ(wij)).2
SetX:=J1∩J2∩J3andY:=X∪K.Bywhatwejustobserved,wij∈WY
foralli,j.Therefore,wij∈WJi∩Jj∩Y.Since|K|≤2byhypothesis,wecanfind
i,j∈{1,2,3}suchthatK∩Ji∩Jj=∅,henceinfactwij∈WX.Weconcludethat
2Inbefactaθ-tthesewistedsubinwvordsolutionareouniquelyccurringasθdetermined-codistance,bythebutthisrequiremenisnottofthatimpateacortancehstephere.theremust

70

R4.6.residues2ank

θItwefollocanwsdirthatectlyR123descend∩Cθ=from∅,cinandtoCviaaccordinglyagalleryR123of=t∅yp.ewijFinally,stayingPropwithinositionR4.4.3123.
impliesthatR123=Rθ123isconnected.
InExample2.1.9(2.1.9)wesawthatKcanbearbitrarylarge,evenforirreducible
flips,buildings.orfindItwcounouldteberexamples.interestingButtoatleastextenditcothevaersbovtheeimpresultortantotcasearbitraryofpropquasi-er
quasi-flips(forwhichK=∅).

residues2Rank4.6.In4.1.7.thisThatsectionis,wewhenstudydirectwhendescenratnkin2toRθresiduesispossiblesatisfyandthewhenprerequisitesRθisofconnected.Theorem
Wewillfocussolelyonthecaseof2-sphericalMoufangtwinbuildings.Themain
reasonisthatfortreeresidues,thereisonlyoneuniqueminimalgallerybetweenany
tworesiduesgivencalonehambcannoters.wHenceorktourhereideainofbgeneral.ypassingHowever,problematic2-sphericalchamtbwinersinsidebuildingsrankare2
alreadywithoutdirectsufficienfactlytorsintofterestingypeA˜ob1).jects(includingallsphericalandallaffinebuildings
RecallthatforθaresidueRofthepositivehalfofthetwinbuilding,theinduced
flip-flopsystemRconsistsofallchambersinRwithminimalnumericalθ-codistance,
is,thatRθ:={c∈R|lθ(c)=d∈minRlθ(d)}.
θthereFexistsurthermoreagalleryrecallinthatRdirtoeactcdeschambenterinintoRθRwithisptheossiblepropifertforyanthatyclθhamisberstrictlyinR
.gallerythatalongdecreasingIntherestofthischapter,Rwillbeanarbitraryrank2residueofthepositive
halfofthetwinbuilding.By2-sphericityitisactuallyaMoufangpolygon.
andIttheturnsinducedouttomapbeθimp:=ortanprojtto◦θonstudyQ.theThepromotivjectionationresidueforthatQis:=theprojRfollo(θ(Rwing))
QalternativecharacterizationofRθ(whichholdsforarbitrarysphericalresiduesR):
Proposition4.6.1(Propositionθ3.5in[DGM]).LetRbeasphericalresidueinC+,
l+set(d,Qθ(:=d))proarjRe(θ(R))simultane.ThenouslyRcmaximal,onsistsofwheralledchamb:=proersjc(c)such.thatbothl+(c,d)and
QProof.LetIbethetypesetofR,letc∈Rbearbitrary.Denotebydtheprojection
ofctoQ.Bydefinition,θ(d)istheprojectionofθ(d)toQ.Moreover,
δ+(c,θ(d))=δ+(c,d)∙δ+(d,θ(d)).

71

4.Structureofflip-flopsystems

Figure4.2.:AsphericalresidueRwithRθandQdepictedforQ=R.

LetwdenotethecodistanceofRandθ(R)(equivalently,thatofQandθ(Q),or
thatofθ(d)andθ(d)).ByrepeatedlyapplyingLemma1.6.5,weget(seealsoFigure
4.2):

δ∗(c,θ(c))=δ+(c,d)∙δ∗(d,θ(c))
=δ+(c,d)∙δ∗(d,θ(d))∙δ−(θ(d),θ(c))
=δ+(c,d)∙δ+(d,θ(d))∙δ∗(θ(d),θ(d))∙δ−(θ(d),θ(c)).
wtheNowθmaximalisaquasi-flipelementofandthetheredoubleforecosetpreservWeIswnWIumerical(cf.[AB08,distances.LemmaMoreo5.148]).ver,wWise
f(c)conclude:=2lthat(c,dlθ)(+c)l=(d,l(θw()d−))2isl+(c,dmaximal.)−l+(d,θ(d)).ThuscisinRθifandonlyif
++canaDenotelwaysthefindacmaximalhambnercumericalwhichdistahasncetheofsameanycprohambjectionerindRtotoQQasbcynhas,.Nobutwwhase
ifcmaximal∈Rθthendistancef(c)=fromn+Q.l(d,Hencθe(dif)).cisinRθ,thenl+(c,d)mustequaln.Thatis,
+bym.DenoteDenotethebymaximalQθthensetumericalofchambdistanceersinofQanywhicchamhθbermoxvinesQfrommaximallyits.imageHenceθ(xthe)
Wmaximalevconcludealuel+tha(td,cθ(∈dR))θifcanandattainonlyisifmf,(c)hence=mf(+c)nisifatandmostonlymif+cnis.oppositeQ
andprojQ(c)isinQθ.Theclaimfollows.

Aswearedealingexclusivelywithrank2residuesR,wedefineanddistinguish
threecases,basedontherankofQ=projR(θ(R)):
(1)Risθ-orthogonaliftherankofQis0,i.e.,Qconsistsofasinglechamber.
(2)Risθ-acuteiftherankofQis1,i.e.,QisapanelofR.

72

ankR4.6.residues2

(3)Risθ-parallelifRisparalleltoθ(R)andthusQ=R.
underInthewhichnextthreeconditionsRsections,θisweconnected,studyeacandhcasewhenindirectdetail.descenSptinecificallytoR,θwisepossible.analyze

4.6.1.Risaθ-orthogonalrank2residue
IfbytheProprankositionofQ=4.6.1,proRjRθ(θ(Requals))theequalsset0c,opthenofcQhambconsistersofoppaositesinglesomechamfixedbercc.hambThener
c.Thissetandthefollowingresultarewell-known:
oppPropositeaositionchamb4.6.2erina(PropMoufangositionp7inolygonis[Abr96];conneseectedalsoif[AandVM99])onlyif.theThepgeolygonometryis
notassociatedtoanyofthegroupsC2(F2)=∼Sp4(F2),G2(F2),G2(F3)or2F4(F2).
Furthermore,wereadilyobservethefollowing:
Proposition4.6.3.IfRisθ-orthogonal,thendirectdescentintoRθispossible.
Proof.ForanychamberdinR,ifdisnotinRθ=cop,hencenotoppositec,then
wePropcanositionfinda4.6.1,chamdberhasdnadjacenumericaltθto-codwhicdistancehisstrictlyfartherawsmallerayfromthancthatthanofddis..ByBy
repeatingthisfinitelymanytimes,wedirectlydescendintoRθ.

4.6.2.Risaθ-acuterank2residue
IfRisθ-acute,thentherankofQ=projR(θ(R))is1,i.e.,Qisapanel.Sincewe
areinaMoufangpolygon,QisendowedwiththestructureofaMoufangset.Recall
thatθinducesanautomorphismoftheMoufangsetQ,namelyθ:=projQ◦θ.
DefineT:=Q\Qθ,thecomplementoftheinducedflip-flopsysteminQ.Equiva-
lently,Tisthecomplementofthesetofelementsmovedmaximallybyθ.Henceif
θ=id|Q,thenTistheemptyset;otherwiseitisthesetofchambersinQfixedby
θ.Ineithercase,byLemma3.3.4,TisapropergeneralizedMoufangsubsetofQ.
Proposition4.6.4.IfRisθ-acute,thendirectdescentintoRθispossible.
Proof.Proposition4.6.1impliesthatthechambersinRθarethosewhichareopposite
theresidueQandwhichareprojectedontoachamberinQθ=Q\T.
DirectdescentintoRθispossiblebyasimilarargumentasinProposition4.6.3:
ForanychambercinR,ifcisnotoppositeT,thenwefindanadjacentchamber
cwhichisfartherawayfromQ,andhencehaslowernumericalθ-codistance.Ifc
isalreadyoppositeQ,thenitiscontainedinapanelPoppositeQ.ThenPmust
containachambercwhichisalsooppositeQandprojectstoachamberinQθ,and
whichhenceisinRθ.

73

4.Structureofflip-flopsystems
WenowturntothequestionwhetherRθisconnectedasachambersystem.For
digonsthisistriviallytrue:Theincidencegraphofthepoint-linegeometryofa
digonisacompletebipartitegraph.Theflip-flopsystemisasubsetofthepoints
andlines(containingelementsofbothtypes,asitconsistsofchambers).Therefore
thegraph,inducthusedincidenceconnected.graphSowefornexttheflip-flopconsidertrsystemiangles.isonceagainacompletebipartite
Proposition4.6.5.IfRisaθ-acuteMoufangprojectiveplane,thenRθisconnected.
Proof.Byinvokingduality,wemayassumethatQcorrespondstoalineK,and
henceTcorrespondstoapropersubsetofthepointrowofK.ByProposition4.6.1,
theelementsofRθareallchambers(p,L)wherepisnotonK,andLmeetsKina
pointqoutsideofT.

Figure4.3.:Connectinggoodchambersinsideaθ-acuteMoufangprojectiveplane.
Let(p1,L1)and(p2,L2)betwosuchchambers.AssumeL1andL2meetinsome
pointr.IfrisapointoutsideK,wehaveaconnectionandaredone.Soassume
r∈K.ByLemma3.3.6,theremustbeasecondpointrinQ\T,differentfromr.
Thenthereexistsachamber(p3,L3)withp3notonKandL3meetingKinr(see
Figure4.3).Bywhatwesaidpreviously,thisnewchamberisconnectedinsideRθto
ourtwooriginalchambers.
Proposition4.6.6.LetRbeaθ-acuteMoufangquadrangleoforder(s,t).Then
Rθisconnectedunlesstheorder(s,t)ofRis(2,2),(2,4),(4,2),(3,3)or(4,4)
(i.e.,associatedtooneofthegroupsC2(Fq)=∼Sp4(Fq)forq∈{2,3,4}or2A3(F2)=∼
PGU4(F2)).
Proof.Byinvokingduality,wemayassumethatQcorrespondstoalineK,andhence
Tcorrespondstoa(proper)subsetofthepointrowofK.ByProposition4.6.1,the
elementsinRθareallchambers(p,L)whereLisoppositeKandpprojectstoa
pointonKoutsideofT.Wecallpointsandlinessatisfyingthesepropertiesgood
pointsandlines.Observethatallbutoneofthelinesinthepencilofagoodpoint
aregoodlines.Moreover,byLemma3.3.6,wehave|K\T|≥2,thereforeevery
goodlinecontainsatleasttwogoodpoints.
Taketwosuchchambers(p1,L1)and(p2,L2).Denotetheprojectionsofp1andp2
toKbyq1andq2,andtheprojectionlinesbyK1andK2.Ifq1andq2areequal,

74

ankR4.6.residues2

useand(thep,Lfact2)arethatL2connectedcontainsinRaθ,secondandpgooprodpjectsointtop2a.poinThentontheKchamdifferenberst(frop2m,Lq21.)
22Henceitsufficestodealwiththecasewhereq1andq2differtoestablishourclaim
.ygeneralitfullin

(a)ProjectingfromKtoL1toK1toobtainT1(b)ProjectingbetweenK1andK2
Figure4.4.:Connectinggoodchambersinsideaθ-acuteMoufangquadrangle.

AgainbyLemma3.3.6,theremustbeasecondpointp1onL1whichdoesnot
projecttoT.TakeanylineL1throughthatpointnotmeetingK.ThenprojectT
toL1andfromtheretoK1,toobtainasetT1of“bad”pointsonK1(seeFigure
4.4a).NotethatallprojectionlinesbetweenL1andK1,exceptfortheonethrough
q1,areoppositeK.AllpointsonK1whicharenotinT1andarenotq1aregood
points,andbyconstructionreachablefrom(p1,L1).
Bysymmetry,wecandothesamewiththesecondchambertoobtainasimilar
subsetT2ofK2.Finally,weprojectfromK1toK2(seeFigure4.4b)andapply
Lemma3.3.7ifthepointorderisatleast5:K1cannotbecoveredbyT1,the
projectionofT2,andthesinglepointq1(whichalsoistheprojectionofq2).This
impliesthattheremustbeaprojectionlinebetweenK1andK2butdifferentfrom
Kwhichmeetsthosetwolinesingoodpoints.ThislineclearlydoesnotmeetK,
henceisgood,andsoweθhaveconstructedasuitableconnectionbetweenourtwo
startingchamberswithinR.
Thisleavesafinitenumberofpotentialexceptionsforquadrangles,namelythe
Moufangquadranglessatisfyings+t≤8.Bycomputercalculations(seeAppendix
A.1),itturnsoutthattheonlycounterexamplesexistinthequadranglesoforder
(2,2),(2,4)(anditsdual),(3,3)and(4,4)–thesearethesmallestexistingMoufang
quadrangles.

ForMoufanghexagons(andpossiblyoctagons),Ibelievethatasimilarstatement
holds,butnogeneralproofisknowntomeatthistime.However,thefollowing
countingargumentprovesconnectednessformostfiniteMoufanghexagons.

75

4.Structureofflip-flopsystems

RθPropisconneositionctedif4.6.7.sLandettRbbotheaarθeat-acuteleastfinite19andMoufangarenothexagondivisibleoforbyder2(ors,t3)..Then
Sketchofproof.Byinvokingduality,wemayassumethatQcorrespondstoalineK,
andhenceTcorrespondstoa(proper)subsetofthepointrowofK.ByProposition
4.6.1,theelementsinRθareallchambers(p,L)whereLisoppositeKandpprojects
toapointonKoutsideofT.Wecallpointsandlinessatisfyingtheseproperties
goodpointsandlines.
ThenumberofgoodlineshenceequalsthenumberoflinesoppositeK,whichis
s2tHence3.LetRθx:=consists|T|−of1.(s−Thexn)s2eact3hgogoooddclihanembconers.tainss+1−|T|=s−xgoodpoints.
Ontheotherhand,startingatagoodlineL,onecanreachatleast
1+(s−x)(t−1)+(s−x)(t−1)(s−x−1)(t−1)
+(t−1)(s−x−1)(t−1)(s−x−1)(t−1)
=t∙1+(s−x−1)(t−1)+(s−x−1)2(t−1)2
θgotimesodthislines.numTber.herefore,DividingthesizethenofumaberofconnectedgoodccomphambonenerstbyofthisRnisumatbleer,astwe(s−obtainx)
anupperboundonthenumberofconnectedcomponents:
22ts#connectedcomponents≤1+(s−x−1)(t−1)+(s−x−1)2(t−1)2.(4.1)
Ifsandtarenotdivisibleby2and3,thenbyLemma3.3.6,wegetx≤5s(as5
is(4.1)theyieldssmallestthatforprimesnandumbteratbiggleaster19,thanthe2nandumb3).erofComconnectedbiningthiscompwithonentsinequalitislessy
.2thanRemark4.6.8.AsimilarcountingapproachcanbeusedforfiniteMoufangquad-
teristicrangles,2.butForyieldsthiswreorseasonbaoundscounthantingPropargumenositiontalso4.6.6,failsandfordoesMoufangnotcooverctagonscharac-as
theseonlyexistincharacteristic2.

4.6.3.Risaθ-parallelrank2residue
Risθ-paralleliftherankofQ=projR(θ(R))is2,i.e.,Q=R.Henceθinducesan
(anti-)automorphismθ:=projR◦θθofR.AsRisaMoufangpolygon,θiseither
theidentity(inwhichcaseR=Randnothinghastobedone),aninvolutory
automorphismoftheunderlyingpoint-linegeometry(mappingpointstopointsand
linestolines),orapolarity(interchangingthetypes).Wewilldealwiththelatter
twocasesseparately.
Inthenextfewpages,wecompletelytreatMoufangprojectiveplanes,andgive
partialresultsonMoufangquadranglesandhexagons.Fordigons,allclaimsaretriv-
ial:Theincidencegraphofthepoint-linegeometryofadigonisacompletebipartite

76

residues2ankR4.6.

graph.Theflip-flopsystemisasubsetofthepointsandlines(containingelements
ofbothtypes,asitconsistsofchambers).Thereforetheinducedincidencegraphfor
theflip-flopsystemisonceagainacompletebipartitegraph,thusconnected.Direct
descentispossiblebecauseanypoint(orline)isincidentwithalllines(points)of
system.flip-floptheθisapolarity
Givenapolarityθ,recallthatanabsoluteelementisapointorlinewhichis
mappedbythepolaritytoanincidentlineorpoint,thatis,xisabsoluteifandonly
ifx∼θ(x).
Remark4.6.9.TheonlyMoufangquadranglesthatadmitpolaritiesaredefinedin
characteristic2,i.e.,withrootgroupswhicharenotuniquely2-divisible;seee.g.
Theorem7.3.2andCorollary7.4.3in[VM98].
ForMoufanghexagons,polaritiesexistonlyforthemixedhexagonsoverfields
admittingaTitsendomorphism,cf.Theorem7.3.4inloc.cit.,whichimpliescharac-
.3teristicFinallybyTheorem7.3.6inloc.cit.Moufangoctagonsdonotadmitpolaritiesat
all.Inviewofthisremark,thefollowingpropositiondealswithcharacteristic2exclu-
.elysivProposition4.6.10.SupposeRisaθ-parallelMoufangquadrangleandθapolarity.
ThenRθconsistsofPhanchambersanddirectdescentispossible.Furthermore,Rθ
isconnectediftheorderofthequadrangleisnot(2,2).
Proof.AnylineLcontainsatmostoneabsolutepoint:Forassumep1andp2inL
areabsolutepoints.Thenθ(p1)andθ(p2)wouldbelinesmeetinginthepointθ(L).
Sincenotrianglesmayexist,weconcludethatθ(L)∈L,andsoLisabsoluteand
p1=p2=θ(L).
Dually,everypointiscontainedinatmostoneabsoluteline.Thus,weseethat
everychamberconsistingoftwoabsoluteelementsisadjacenttoachamberwithonly
oneabsoluteelement,andanysuchchamberinturnisadjacenttoachamberwith
twonon-absoluteelements.Finally,achamberconsistingofanon-absolutepoint
andanon-absolutelineismappedbyθtoanoppositechamber.Thisprovesthat
directdescentispossibleandthatRθconsistsofPhanchambers.
Giventwochambers(p1,L1)and(p2,L2)inRθ,wewanttoconstructaconnection
insideRθ.Ifp1∈L2orp2∈L1,wearedone.Ifthatisnotthecase,wemayupto
dualityassumethatL1isoppositeL2.
Iftheorderofthequadrangleis(2,2),thenthereisapolarityforwhichRθconsists
oftwoconnectedcomponents,eachformingapentagon.3Thispolaritycanbeeasily
understoodbylookingatFigure4.5:Itinterchangeseachouter“corner”withthe
3Infactthisistrueforall36polarities,astheyareallconjugate,butwedonotneedthishere.

77

4.Structureofflip-flopsystems

oppositeouteredge;eachpointonthemiddleofanouteredgeismappedtotheline
spannedbyitandtheoppositecorner;andhenceeachinnerpointisinterchanged
withthe“curved”linepartiallyencirclingit.Oneconnectedcomponentcontains
allchamberswhichconsistsofa“corner”andanouteredge;theothercontainsthe
chambersmadefrominnerpointsandthe“curved”linesbetweenthem.

Figure4.5.:Collinearitygraphofthegeneralizedquadrangleoforder(2,2).

Assumenowthattheorderisatleast(3,3).Everylinecontainsatmostone
absolutepoint,henceL1andL2eachcontainatleastthreenon-absolutepoints.
Hencewecanfindtwogoodpointsa1andb1onL1whichprojecttogoodpoints
a2andb2onL2.Ifoneoftheprojectionlinesisnon-absolute,wearedone.So
assumethatbotha1,a2andb1,b2areabsolute.Takeasecondnon-absoluteline
Lpro3jectthrougha2andtheb2thirdtoL3go.odBothpointproc1onjectionL1,linesonemwhicusthbedogoesodnot(asintheytersectareL2.differenThent
fromtheuniqueabsolutelinesthrougha2andb2).Atmostoneofa2andb2can
projecttoanabsolutepointonL3(asitcontainsatmostone);henceweobtaina
connection.dogoWhatremainsisthecaseofMoufangprojectiveplanes,wherepolaritiesoccur
.tifullyplenProposition4.6.11.SupposeRisaθ-parallelMoufangprojectiveplaneandθa
polarity.ThenRθconsistsofPhanchambers.Moreover,itisconnectediftheplane
isdifferentfromP2(F4),i.e.,notassociatedtothegroupA2(F4)=∼SL3(F4).
θPr.oof.AssumeFirstwthereeareestablishnotheabsolutepexistenceoints,ofahencechamnoberabsoluteinRopplines.ositeThentoitseveryimagechamunderber
ismappedtoanoppositeone.Soassumethereisanabsolutepointp.Thenexactly
ponewhiclinehLarenotthroughpabsolute.isabsolute,IftherenamelyisaL=θ(non-absolutep).SopthereointpareonlinesL,L,theLchamthroughber
(p,L)ismappedtoanoppositeone.IfallpointsonLareabsolute,theninfact
allitsabsoluteabsoluteplineointswmouldustbmeeteonLLina(iftheresecondwasabsoluteanpabsoluteoint,pwhicoinhtpisimpoutsideossibleL,sincethen
absolutelinescontainexactlyoneabsolutepoint).Hencewefindanon-absolute
pointponLandareagaindonebychoosingthechamber(p,L).

78

residues2ankR4.6.

Taketwochambers(θp1,L1)and(p2,L2)inRθ.Iftheyarenotopposite,θthenthey
areconnectedinsideR.E.g.ifp1isonL2,then(p1,L2)isachamberinR.
Hencewemayassumethat(p1,L1)and(p2,L2)areopposite.Assumefurthermore
thattheorderoftheplaneisatleast5.Thesetofabsolutelinesthroughp1forms
apropergeneralizedMoufangsubsetofthepencilofp1(toseethis,justapplyθ
tothepencil,thenprojectback,anduseLemma3.3.4);sodoesthesetofabsolute
pointsonL2.Projectingthepencilofp1toL2andapplyingLemma3.3.7,there
mustbeanon-absolutelinethroughp1whichmeetsL2inanon-absolutepoint,and
wehavethedesiredconnection.
ForP2(F2),wecanusethesameargument,sincebycountingweseethatthe
propergeneralizedMoufangsubsetmusthavesizeone(anontrivialinvolutionona
setofthreeelementshasonefixedpoint).ForP2(F3)wegiveacomputerproofin
A.2.endixAppRemark4.6.12.ForP2(F4),thereisacounterexample.Namely,takeahermitian
resp.unitarypolarityofF43.ThenRθconsistsoffourconnectedcomponents,each
containingsixchambers(moreprecisely,eachcomponentcontainsthreepointsand
threelinesformingacompletebipartitegraph,cf.[BS04,Section2]).
WenextconsiderthedirectdescentpropertyinMoufangprojectiveplanes.Here,
[Bae46]originallyinspiredustoproveProposition2.5.2,whichreadilyimpliesthe
following(whichcorrespondstoCorollary2inloc.cit.):
Lemma4.6.13.LetθbeapolarityofaMoufangprojectiveplaneRdefinedover
analternativedivisionalgebraA.Ifθadmitsanon-absolutelineLsuchthatall
pointsonLareabsolute,thenthecharacteristicofAis2.
Proof.AssumethereexistsalineLwithallpointsonitbeingabsolute.Pickany
suchpointpandthenpickanon-absolutelineKthroughitdifferentfromL.We
induceanautomorphismϕonthepointsetofKbycomposingθwiththeprojection
map,i.e.,ϕ(x):=projK(θ(x))forallx∈K.Clearlyϕfixesonlythepointp.Butby
Proposition2.5.2thismeansthattheadditivegroupofAisnotuniquely2-divisible.
HencethecharacteristicofAis2.
Wecannowdeducethedesiredresultincharacteristicdifferentfrom2.Incharac-
teristic2,therearecounterexamples,seeRemark4.6.15.Accordingly,thisisthebest
wecanhopeforwithoutimposingfurtherrestrictionsontheplaneorthepolarity.
Proposition4.6.14.SupposeRisaθ-parallelMoufangprojectiveplaneandθa
polarity.Thendirectdescentispossibleiftherootgroupsareuniquely2-divisible.
Proof.Let(p,L)beachamberinR.Ifθ(p)=L,i.e.,thechamberisfixedbyθ,
thenforanyotherlineLthroughp,thechamber(p,L)isnotfixed,andhencethe
reduced.isdistance-coθAssumenextthatpisanabsolutepoint,butLisanon-absoluteline.ByLemma
4.6.13,theremustbeanon-absolutepointponL.Therefore,thechamber(p,L)
ismappedtoanoppositechamber,andwearriveinRθ.

79

4.Structureofflip-flopsystems

Remark4.6.15.By[Bae46],incharacteristic2,therearepolaritiesoffiniteDe-
sarguesianprojectiveplanesforwhichallabsolutepointsarecollinear,andinfact,
formthepointrowofasinglelineL.ByAppendixAinloc.cit.,thisisinparticular
alwaysthecaseiftheorderofthefieldisnotasquare.Inthisscenario,thechamber
(p,L)(foranyp∈L),consistingofanabsolutepointandanon-absoluteline,is
neitherfixednormappedtoanoppositechamber(sonotcontainedinRθ).Butall
adjacentchambersalsocontainanabsolutepoint,hencearenotcontainedinRθ.So
directdescentisimpossible.
Remark4.6.16.WehavenotdealtwithMoufanghexagonsandoctagons.However,
wehavestrongreasonstobelievethatconnectednessanddirectdescentholdatleast
forhexagons.ThisissubjectofongoingresearchbyHendrikVanMaldeghemand
[HVM].authorthe

θisaninvolutoryautomorphism
Proposition4.6.17.SupposeRisaθ-parallelMoufangprojectiveplaneandθan
involutoryautomorphism.ThenRθisconnectedanddirectdescentispossible.
Proof.Sinceθisnottheidentity,theremustbealineLmovedbyθtoadifferent
line.AnysuchlineLcontainsauniquefixedpoint,namelythepointwhereLand
θ(L)meet.Likewise,anynon-fixedpointisonauniquefixedline.Consequently,
adjacenttoafixedchamberwealwaysfindachamberconsistingofonefixedandone
bothnon-fixedelementselemenaret;noandn-fixed,adjacenti.e.,toasucchhamabcerhambwhicer,hwisealwmappaysedfindtoaanchamoppbeositerwhereone.
ThusRθcontainsallchambersmappedtoanoppositechamber,anddirectdescent
ossible.pisToseethatRθisconnected,let(p1,L1)and(p2,L2)betwochambersinRθand
considerthesetofalllinesfromp1topointsonL2.Exactlyoneoftheseisfixedby
θ,andexactlyonemeetstheuniquefixedpointonL2.Henceanyoftheremaining
linesthroughp1isnon-fixedandmeetsL2inanon-fixedpoint,andwearedone.
Forquadrangles,wecancurrentlyonlydealwiththeclassicalquadranglesusing
theresultsfromSection3.3.ThiscoversallfiniteMoufangquadrangles.
Proposition4.6.18.SupposeRisaθ-parallelclassicalquadrangleandθanin-
volutoryautomorphism.ThenRθisconnectedunlesstheorder(s,t)ofRis(2,2),
(2,4),(4,2),(3,3)2or(4,4)∼(i.e.,associatedtooneofthegroupsC2(Fq)=∼Sp4(Fq)
forq∈{2,3,4}orA3(F2)=PGU4(F2)).
Proof.InSection3.3.4weprovedthiswhenthesizeoftheunderlyingfieldKis
biggerthan9anddifferentfrom16,inparticularforinfinitequadrangles.2
s∈{This2,3,le4a}v;esandthe(s2,follos3)wingfors∈orders:{2,3,(4s,}.s)forTheses∈are{2,dealt3,4,5with,7,in8,9,App16};endix(s,sA.3)bfory
computations.hinemac

80

residues2ankR4.6.

Weconjecturethefollowingresulttoholdingeneral:
Conjecture4.6.19.SupposeRisaθ-parallelMoufangquadrangleandθanin-
volutoryautomorphism.ThenRθisconnectedunlesstheorder(s,t)ofRis(2,2),
(2,4),(4,2),(3,3)2or(4,4)(i.e.,associatedtooneofthegroupsC2(Fq)=∼Sp4(Fq)
forq∈{2,3,4}orA3(F2)=∼PGU4(F2)).
AswithMoufangprojectiveplanes,wecanprovethatdirectdescentispossible
ifthecharacteristicisdifferentfrom2.Unlikethere,however,wecurrentlyarenot
awareofanyactualcounterexamples,sothismightsimplybeduetoourproofbeing
t.deficienProposition4.6.20.SupposeRisaθ-parallelclassicalquadrangleandθaninvo-
lutoryautomorphism.Thendirectdescentispossibleiftherootgroupsareuniquely
-divisible.2Proof.Fordirectdescentinquadrangles,twothingscangowrong:
(1)Theremightbeachamber(p,L)fixedbyθsuchthatalladjacentchambers
arefixedbyθaswell.Butthenθisaµ-map,i.e.,theproductoftworoot
elations,andweareincharacteristic2,asonlythenµ-mapscanbeinvolutory.
Toseethatθisaµ-map,chooseasecondfixedpointqonL,andasecond
fixedlineMthroughp.ThenchoosealineKthroughqdifferentfromLand
apointronMdifferentfromp.Now,(K,L,M)definesauniquerootofthe
quadrangle,andthereisauniquerootelationφassociatedtothatrootwhich
mapsrtoθ(r)∈θ(M)=M.Beingarootelation,italsofixesthepointrow
ofLandthepencilsofqandp.Likewise,thereisauniquerootelationψ
associatedtotheroot(q,p,θ(r))whichsendsthelineKtoθ(K)θ(q)=q,
andwhichfixesthepencilofpandthepointrowsofLandM.Accordingly,
ψ◦φfixespanditspencil,Landitspointrow,andsendsrtoθ(r)andK
toθ(K).But(q,K)and(r,M)areopposite,hencedefineanapartment,on
whichθandψ◦φcoincide;butthetwomapsalsocoincideonallchambers
adjacentto(p,L).Forthisreasonandbyrigidityofthicksphericalbuildings
(seee.g.[AB08,Corollary5.206])),θequalsψ◦φ.
(2)Uptoduality,therecouldbeagoodlineL(i.e.,Lisnotincidenttoθ(L))
suchthatallpointsonitarebad(collineartotheirimage).Henceeachpoint
ponLisonauniqueθ-fixedlineLp.Then(p,L)isachamberwithnumerical
θ-codistance3,andallchambersadjacenttoithavenumericalθ-codistanceat
most3,sowecannotdescendfurther.ButinPropositions3.3.9and3.3.11
weprovedthatifthissituationoccursinaclassicalquadrangle(definedin
characteristicdifferentfrom2),thenallpointsarecollineartotheirimage,
hencethemaximalnumericalθ-codistanceis3.
Again,westronglybelievethattheaboveholdsinthegeneralcase:

81

4.Structureofflip-flopsystems

Conjecture4.6.21.SupposeRisaθ-parallelMoufangquadrangleandθaninvo-
lutoryautomorphism.Thendirectdescentispossibleiftherootgroupsareuniquely
-divisible.2

ThequestionofwhetherRθisconnectedforMoufanghexagons,orwhetherdirect
descentispossible,isstillopen.Dealingwiththeseandtheremaining(exceptional)
quadranglesissubjectofongoingresearchbyHendrikVanMaldeghemandtheauthor
[HVM].Forclassicalhexagons(i.e.,splitCayleyhexagons),somepromisingpartial
resultsalreadyhavebeenachieved.
Finally,nothingisknowntousregardingMoufangoctagons,butsincetheseonly
occurincharacteristic2,wedonotlosetoomuch(aswehavetoexcludecharacteristic
2inmanyotherθplacesanyway).Nevertheless,itwouldbeinterestingtoatleast
knowwhetherRisconnectedforoctagons,aswethencouldapplythistostrong
quasi-flips,wheredirectdescentisalwayspossible.

4.7.Statementofthemaintheorems
Combiningallwehavedoneintheprecedingsection,wearriveatthefollowingmain
theorems:Theorem4.7.1(jointworkwithVanMaldeghem).LetRbeaMoufangprojective
planeoforderdifferentfrom4,oraclassicalquadrangleoforder(s,t),st>16.
LetθbeaninvolutoryautomorphismorapolarityofR.ThenRθisconnected.θIf
furthermoretherootgroupsareuniquely2-divisible,thendirectdescentintoRis
ossible.pAsanimmediateconsequence,weobtainageneralizationofTheorem4.1.8:
Theorem4.7.2(jointworkwithGramlichandMühlherr).Letθbeaquasi-flipof
anRGD-system(G,{Uα}α∈Φ,T)oftype(W,S),whereallrootgroupsUαareuniquely
2-divisible.Assumeallrank2residuesoftheassociatedtwinbuildingCareprojective
planes,orclassicalquadranglesoforder(s,t),st>16.
Thenforallrank2residuesR,directdescentintoRθispossibleandRθiscon-
d.cteneCombiningthiswithTheorem4.1.7yieldsthefollowingversionofTheorem4.1.10:
Theorem4.7.3(jointworkwithGramlichandMühlherr).Letθbeaquasi-flipof
anRGD-system(G,{Uα}α∈Φ,T)oftype(W,S),whereallrootgroupsUαareuniquely
2-divisible.Assumeallrank2residuesoftheassociatedtwinbuildingCareprojective
planes,Thenortheclassicflip-flopalquadrsystemanglesCθisofcoronnedercte(ds,t)and,est>quals16.theunionofallminimalPhan
residueswhichθinturnallhaveidenticalsphericaltypeK.Thechambersystemof
K-residuesofCisconnectedandresiduallyconnected.

82

4.7.Statementofthemaintheorems

Asalreadystatedpreviously,Ibelievethattheabovecanbeextendedtoalsocover
residuesisomorphictosplithexagons(thusextendingtheresultsabovetoalmostall
splitgroupsincharacteristicdifferentfrom2).

Finally,wecontrastthiswiththefollowingbeautifulresultonstrongflips,due
toDevillersandMühlherr.Byrestrictingtostrongflips,manyofthetechnical
complicationswehadtodealwithcanbeavoided(forexample,directdescenθtis
rankimplied2byconnectednessLemma2.4.2)(as.weThisdid,enablesalthoughthemwetoalsodeduceexplicitlyconnectednesscomputedofwhenCfromthis
localconnectednessholds).Butmoreover,theycanconcludesimpleconnectedness
residues.3rankstudyingfrom

Theorem4.7.4(Proposition6.6in[DM07]).LetCbeaMoufangtwinbuilding,let
θbeastrongflipofC.Supposethefollowingconditionsaresatisfied:
(1)Cis3-spherical(ifJ⊂Sisofcardinalityatmost3,thenJisspherical).
(2)Forallrank2residuesRofC+,thechambersystemRθisconnected.
(3)Forallrank3residuesRofC+,thechambersystemRθissimplyconnected.
ThenCθissimplyconnected.

83

4.

84

uctureStr

of

flip-flop

systems

CHAPTERFIVE

TRANSITIVEACTIONSONFLIP-FLOPSYSTEMS

LetGbeagroupwithtwinBN-pair,letθbeaquasi-flipofG.Inthischapter
westudytransitivitypropertiesoftheactionofGθ,thecentralizerofθinG,on
thebuildingandontheflip-flopsystemCθasdefinedinChapter4.Ourmotivation
fordoingsoisthatgivenasufficiently“nice”transitiveactionofourgroupG,we
canderivemanyinterestingpropertiesfromthis.Forexample,presentationsof
thecentralizerGθofθ(Theorem5.4.2),generalizedIwasawadecompositions(cf.
Theorem5.4.7),latticesinKac-Moodygroups(cf.Theorem6.2.7duetoGramlich
andMühlherr)orfinitegenerationofGθ(cf.Theorem6.2.5inChapter6).
TheworkpresentedinthischapterispartiallybasedonjointworkwithTomDe
MedtsandRalfGramlichin[DMGH09],specificallythepartsonrank1transitivity
inSection5.3andonIwasawadecompositionsinSection5.4.Theresultsonlattices
areduetoBernhardMühlherrandRalfGramlich[GM08].

yransitivitT5.1.ofInthiselementssection,inGθisfixedabyquasi-flipθ.RofecallafromgroupGPropwithositiontwinB2.2.1Nthat-pair,θandinducesGθisathebuildinggroup
quasi-fliponthetwinbuildingCassociatedtoG,whichwealsodenotebyθ.
Definition5.1.1.Forw∈W,set
Cwθ:={c∈C+|δθ(c)=w},
whereδθ(c):=δ∗(c,θ(c))asdefinedinSection2.1.
Sincetheθ-codistanceofachamberisunique,weobtainapartitionofthepositive
building:theofhalfC+=Cwθ.(5.1)
W∈w

85

5.Transitiveactionsonflip-flopsystems

Remark5.1.2.WhilewedefinedtheCwθtobesubsetsofthepositivehalfofthe
twinbuilding,wecouldjustaswellhavedefinedthemassubsetsofthenegativehalf.
Eitherway,weobtainessentiallyidenticalresults,asθisabijectionbetweenboth
halvθeswhichrespectsthepartitiongivenabove(uptoarelabeling,asδθ(θ(c))=
θ(δ(c))),andallotherpropertieswewillbeinterestedinlater.
Forpracticalpurposes,weareonlyinterestedinthoseCwθwhicharenonempty.
Thismotivatesthefollowingdefinition.
Definition5.1.3.Denotethesetofallθ-codistancesby
Invθ(C):={w∈W|thereexistsachamberc∈Csuchthatδθ(c)=w}.

θθinInLemmaotherw2.3.1,ords,allww∈∈InvInv(θC()C)ifareandθ-tonlywistedifCwinvisolutions,nonempti.e.,y.θ(Asw)we=hawv−e1seanden
Invθ(C)⊆Invθ(W).
θRemark5.1.4.Foraproperquasi-flip(i.eθ.,aquasi-flipadmittingaPhanchamber),
C1Wcoincideswiththeflip-flopθsystemCdefinedinChapter4.Ingeneral,theθflip-
TheflopcasesystemiswherethethereunionisofathoseuniqueCwforwsucwhichhl(thatw)Cisθ=minimalCθisofamongparticallwular∈Ininvte(Wrest.).
wCurrently,weknownoexamplewherethisisnotthecase.
TheactionofGonthebuilding(obtainedbyrestrictingtheactionofG)preserves
thedecompositionθgivenin(5.1),andthusinducesanactiononeachCwθsincefor
,Ggθ∈δθ(g.c)=δ∗(g.c,θ(g.c))=δ∗(g.c,g.θ(c))=δ∗(c,θ(c))=δθ(c).
plainedBythisintheandinthetroprecductionedingofthisremark,cGhapter,θalsoweactsareinontheterestedflip-floinptransitivitsystem.yAspropwasertiesex-
ofthisaction,motivatingthefollowingdefinition.
Definition5.1.5.Wecallθflip-floptransitiveifGactstransitivelyonCθ.We
callθdistancetransitiveifforeachw∈WthegroupθGθactstransitivelyonCwθ.
Finally,wecallθbuildingtransitiveifthegroupGθactstransitivelyonC+.
Examples5.1.6.LetFbeafieldwithafieldautomorphismσoforderatmost2.
LetmatricesG=inSL2G(,F),andletletB+θberesp.theB−σ-tbewistedtheChevsubgroupsalleyofinvupperolution,resp.i.e.,lowtheermaptriangularx→
(tx−1)σ.InthenotationofSection3.1,θ=θ−1,σ.
(a)IfknoFwn=CIwasaandwaσisdecompcomplexosition(seeconjugation,[Iwa49],thenorGθmost=boSU2oks(C)on,Lieandbygroups,thewsucell-h
as[Hel78]or[Kna02]),G=GθB+=GθB−(cf.Corollary5.4.6).Hencethis
e.transitivbuildingisflip

86

yransitivitT5.1.

(b)Letqbeaprimepower.IfFisqthefinitefieldoforderq2,andσitsunique
involutoryautomorphismx→x,thenbyProposition5.3.6,itisflip-flop
transitive.ButbyCorollary5.4.6,itisnotbuildingtransitive.
(c)If(F,σ)=(Q,id),thenGθ=SO2(Q)andθisfarfrombeingbuildingoreven
flip-floptransitive.ThisfollowsfromProposition5.3.4,asinQthesumoftwo
squaresisnotgenerallyasquare.Seealso[HW93,Examples4.12and6.12].
IfCθisthedisjointunionofseveralsetsCwθ,thereisnochanceGθcouldact
transitivelyonit.Ontheotherhand,ifCθequalsoneoftheCwθ,thendistance
transitivitytriviallyimpliesflip-floptransitivity.Westudysomecasesoftheconverse
question(i.e.,findingconditionsunderwhichflip-floptransitivityimpliesdistance
5.5.Sectioniny)transitivitRemark5.1.7.OnemayaskwhenCθ=Cwθ.InChapter4westudiedconditions
underwhichaquasi-flipθisK-homogeneous,i.e.,whenallminimalPhanresidues
haveidenticaltypeK.ThenbyLemma4.3.5,Kissphericalandδθisconstant
andequaltowKonanyminimalPhanresidueoftypeK.Consequently,foraK-
homogeneousquasi-flip,theflip-flopsystemCθequalsCwθK(andalsoisaunionof
-residues).KRemark5.1.8.Inviewoftheprecedingremark,anotherkindoftransitivitycomes
tomind:ForaK-homogeneousflip,itwouldalsoθbeinterestingtoknowwhetherGθ
istransitiveontheK-residuechambersystemCK(cf.Definition4.5.1).Howeverwith
theexceptionofabriefobservationinSection6.1.3wedonotstudythisquestionin
thesis.tpresentheSupposenowGθactsdistancetransitively,andletBbeaBorelsubgroupofG
stabilizingsomechamberc∈C+.Foreachw∈Invθ(W)choosearepresentative
gw∈Gwiththepropertythatgw.c∈Cwθ.Thenbydistancetransitivitywehave
C+=w∈Invθ(W)Gθgw.c.
Andthusonthegrouplevel,usingthatC+=∼G/B,
G=GθgwB.(5.2)
w∈Invθ(W)
Remark5.1.9.ThiscanbeconsideredasaspecialcaseoftheSpringerparame-
terizationofthedoublecosetdecompositiongiveninProposition2.7.5.Wesketch
thisrelationundertheassumptionthateverychamberiscontainedinaθ-stable
twinapartment(cf.Section2.5).Fixaθ-stabletwinapartmentΣcontainingc,and
assumethatthegwwerechosensothatgw.Σisagainθ-stable(inviewofstrongtran-
sitivityofGandtheassumptionthateverychamberiscontainedinaθ-stableapart-
ment,thisiscertainlypossible).Thenθ(gw.Σ)=gw.Σ,thereforegw−1θ(gw)∈NG(Σ).
Thisshouldmakethecorrespondencebetweenthetwodecompositionsclear.

87

5.Transitiveactionsonflip-flopsystems

tenceTheofθdecomp-stabletositionwfromin-apartmenEquationts.The(5.2)wassumptionorkswithoutthatGθtheactsassumptiondistanceontransitivtheexis-ely
eaccanhCalsoθ.beFinallydropp,ed,observbutethen(againwemawithoutyhavefurthertotakemrestrictionsultipleongrepresen)thattativesgwfor
www=δθ(gw.c)=δ∗(gw.c,θ(gw.c))=δ∗(c,gw−1θ(gw).θ(c)).
∗θθ1−gSo,−1θin(gwa)w∈ayN,Ggw(Σ)θ,(gwthen)encowedescanhomakwemucthishδ(gobservw.c)ationdiffersprecise,fromδwhic(c)h=isδex(c,actlyθ(c))ho.wIf
woneobtains,geometrically,thestatementofProposition2.7.5.
Remark5.1.10.InthecaseofF-rationalpointsG(F)ofanalgebraicgroup,resp.
fortionthe6.1.3),assobyciated[HW93,sphericalPropositionbuilding,4.11]and(seeanFalso-linearSectionquasi-flip6.1.3),θall(intheminimalsenseθ-splitSec-
GθparabactsolicFtransitiv-subgroupselyonaretheGθflip-flop-conjugatesystemoverinthethealgebraicbuildingofclosureG(F)F.ofInF.Tparticular,hatis,
thequasi-flipishomogeneousinthesenseofChapter4.Inthesmallerbuilding
associatedtoG(F),flip-floptransitivitymaynolongerhold,howeverhomogeneity
inherited.is

Inspiredbytheproofof[GW,Theorem7.1]wenowdetermineaboundonthe
numberofGθ-orbitsonagivenCwθ,w∈Invθ(W).Namely,undertheassumptionthat
thereeverychamberiscontainedinaθ-stabletwinapartment,onecanobtaingood
initialboundsbystudyingwhathappensinasingleθ-stabletorus(i.e.,asubgroup
conjugatetoB+∩B−andstabilizedbyθ).
Forthis,weneedtointroducesomenotation.
Definition5.1.11.Toeveryquasi-flipθofagroupG,wecanassignthetwist
mapτθ:G→G:g→θ(g−1)g(notethatthisisingeneralnotahomomorphism).
Moreover,theθ-twistedactionofGonitselfisgivenbyy∗τg:=θ(g)−1yg.A
θ-twistedT-orbitistheorbitofagroupTundertheθ-twistedaction.
Lemma5.1.12.Letθbeaquasi-flipofanRGD-system(G,{Uα}α∈Φ,T),letCbe
theassociatedtwinbuilding,andsupposethateverychamberiscontainedinaθ-
stabletwinapartment.Thenforeachw∈Invθ(W),thereexistsa∈Gsuchthat
eachGθ-orbitonCwθcorrespondstoauniquetwistedaTa−1-orbitonτθ(G)∩aTa−1.
Inparticular,ifTisfinite,thereareonlyfinitelymanyGθ-orbitsoneveryCwθ.
Proof.Chooseanyw∈Invθ(W)andanychamberc∈Cwθ.ByTheorem2.5.8,there
existsaθ-stabletwinapartmentΣcontainingc.ThestabilizerTofthepair(c,Σ)
isconjugatetoTsoT=aTa−1fora∈G.
Letc∈Cwθbearbitrary,andchooseaθ-stabletwinapartmentΣcontainingc.
BystrongtransitivityofG,thereexistsg∈Gsuchthatg.c=candg.Σ=Σ.
Thereforeg.θ(c)∈Σ,and
w=δ∗(c,θ(c))=δ∗(c,θ(c))=δ∗(g.c,g.θ(c))=δ∗(c,g.θ(c)).

88

yransitivitT5.1.

SincethereisauniquechamberinΣatanygivencodistancefromc,weconclude
thatg.θ(c)=θ(c).Settingx:=τθ(g)=θ(g−1)g,wecompute
x.c=θ(g−1)g.c=θ(g−1).c=θ(g−1.θ(c))=θ(θ(c))=c,
andsimilarlyx.θ(c)=θ(c)andx.Σ=Σ,thusx∈T.
Thegroupelementgchosenaboveisofcoursenotunique.Indeed,foranyt∈T
wehavegt.c=c=g.candgt.Σ=Σ=g.Σ,andtheseareallelementsofGwith
thisproperty.Wehaveτ(g)∈τ(G)∩T.
Letc∈CwθbeanotherchamberinCwθ,containedinaθ-stabletwinapartmentΣ
andassumehwaschosenfor(c,Σ)asgwaschosenabove.
ByLemma2.7.1),candcareinthesameGθ-orbitifandonlyif(c,Σ)=
gT.(c,Σ)and(c,Σ)=hT.(c,Σ)areinthesameGθ-orbit.Thisisthecaseifand
onlyifhTg−1∩Gθ=∅,ifandonlyif1∈τθ(hTg−1).Thisisinturnequivalentto
τθ(g)∈τθ(hT)=τθ(h)∗τT.
HencecandcareinthesameGθorbitifandonlyiftheθ-twistedorbitsτθ(h)∗τT
andτθ(g)∗τTareequal.
Thislemmahasvarioususefulapplications.WewilluseitinChapter6toprove
acriteriononwhenGθisfinitelygenerated,seeTheorem6.2.5.Itcanalsobeused
toextendtheresultfrom[GM08],whichstatesconditionswhenthecentralizerGθ
ofastrongflipθofalocallyfiniteKac-Moodygroupisalatticeinthecompletionof
theambientKac-Moodygroup,toarbitraryproperquasi-flips.SeeTheorem6.2.7.
Weconcludethissectionwithsomeobservationsonbuildingtransitivequasi-flips.
SinceGθpreservesthepartitionofthebuildinggivenin5.1,buildingtransitivity
impliesthatthereexistsw∈WsuchthatC+=CwθandhencealsoCwθ=Cθ.Accord-
inglybuildingtransitivityimpliesflip-floptransitivityanddistancetransitivity.
AnotherniceconsequenceofbuildingtransitivityisthatitimpliesthattheWeyl
groupiscentralized,atleastifthequasi-flipisproper,asthefollowinglemmashows.
Lemma5.1.13.Supposeθisaquasi-flipforwhichtheθ-codistanceisconstant.If
θisproper,thenitisaflip,i.e.,itcentralizestheWeylgroup.
Proof.Byhypothesis,wehaveδθ(c)=1Wforallchambersc.Letc,dbetwo
arbitrarys-adjacentchambersforsomes∈S.Thenθ(c)andθ(d)areθ(s)-adjacent.
Sinceδ∗(c,θ(c))=1W=δ∗(d,θ(d)),Axiom(Tw2)impliess=θ(s).
Thattheprecedinglemmaonlyappliestoproperquasi-flipsisnotactuallyasevere
restriction,asthefollowinglemmaillustrates.
Lemma5.1.14.Supposeθisaquasi-flipofatwinbuildingCoftype(W,S).If
C+=Cwθforsomew∈W(equivalently,ifδθisconstant),thenthereexistsaθ-stable
sphericalsubsetK⊆Ssuchthatw=wK.Moreover,θinducestheidentityonS\K
andallgeneratorsinKcommutewithallgeneratorsinS\K.

89

5.Transitiveactionsonflip-flopsystems

Proof.ByLemma4.3.5,weknowthatthereisaθ-stablesphericalsubsetKofSsuch
thatwequalsthelongestwordofK.Picks∈Sandanychamberc.ByAxiom(Tw3)
thereisachamberdwhichiss-adjacenttocsuchthatδ∗(d,θ(c))=sw=w.Thus
δθ(d)∈{sw,swθ(s)}.Butbyhypothesis,δθ(d)=w.Weconcludethatswθ(s)=w,
orequivalentlyθ(s)=w−1swforalls∈S.
Supposenowthats∈/K.Sinceθ(s)=wsw=wKswK,theExchangecondition
(resp.theDeletioncondition)andthefactthatKandhenceS\Kareθ-stable,
impliesthatθ(s)=s,andsosw=ws.
Forallv∈WK,wehavev≤wK=wintheBruhatorder.Thusclearlysvs≤
sws=w(seee.g.[Hum90,Proposition5.9]anditsproof).Consequently,sws∈
WK,andWKisnormalinW.ThisimpliesthatKandSbelongtotwodistinct
componentsofthediagramof(W,S).
Remark5.1.15.InviewofLemma5.1.14,irreducibilityisnecessaryinLemma
5.1.14.Otherwise,weeasilyproducecounterexamples:Takeasphericaltwinbuilding
Cadmittingaproperquasi-flipθforwhichallchambersarePhanchambers.Then
taketheproductC×Cofthetwinbuildingwithitself,anddefineaquasi-flipθ:=
θ×(−id)ontheresult(whereby(−id)wemeanthemapwhichinterchangesa
chamberc+withits“twin”c−(cf.Example1.6.8).Clearlyθcannotbeproper,yet
δθisconstantandequalto1W×w0,wherew0isthelongestelementoftheWeyl
.ofgroupCInfact,lookingattheproofoftheLemma5.1.14,thisisallthatcanhappen:If
C+=CwθK,thenourbuildingsplitsintotwodirectfactors:Onesphericalfactorof
typeK,onwhichourquasi-fliprestrictstoa“trivial”quasi-flip(comingfromthe
identity,asinthepreviousparagraph),andonefactoronwhichwegetaproper
buildingquasi-flip,whichthenisaflipbyLemma5.1.13.

5.2.Alocalcriterionfortransitivity
Itonisawell-knoconnectedwncandhambereasytosystemseeCthatoveranIistransitivadjacency-preservingeifandonlyactionifofthereagroupexistsGa
chamberc∈Csuchthatforeachi∈IthenormalizerStabG(Pi(c))actstransitively
onthei-panelPi(c)ofCcontainingc.Forcompleteness,weprovideashortproof
ertheless.nevProposition5.2.1.LetCbeaconnectedchambersystemandletGbeagroupof
automorphismsofC.ThenGactstransitivelyonCifandonlyifthereexistsa
chamberc∈CsuchthatforeachpanelPcontainingcthestabilizerStabG(P)acts
.PonansitivelytrProof.Itsufficestoshowthatforanychamberdthereexistsg∈Gmappingctod.
cSince=dCisfromcconnected,tod.wWeecanprovefindtheaclaimminimalbygalleryinductionc=con0∼ni.1cF1or∼in2c=2∙∙0,∙cwne−1ha∼ivne
nc=dandnothinghastobeshown.Else,assumeweknowtheclaimholdsforn−1.

90

5.3.Transitivityinrank1

1−Thenundertherethismap.existsg0Then∈Gc=gmapping−1cn−1c∼toicng−−11.d=Letdd.:=Byg0hypdbeothesis,thepreStabGimage(Pi(ofc))d
n0n0gacts:=g0g1transitivmapselycontoPdinas(c)g,.cso=(thereg0g1).cexists=gg10.(∈g1.cStab)=G(gP0i.dn(c=))d.mappingctod.Hence

Corollary5.2.2.LetGbeagroupwithatwinBN-pair,letCbetheassociatedtwin
building.Supposeθisaquasi-flipofGandGθthegroupofallelementsfixedbyθ.
ThenGθactstransitivelyonthepositive(resp.negative)halfofCifandonlyifthere
existsachambercsuchthatforeachpanelPcontainingcthestabilizerStabGθ(P)
actstransitivelyonP.

Corollary5.2.3.LetGbeagroupwithatwinBN-pair,letCbetheassociatedtwin
building.Supposeθisaquasi-flipofGandGθthegroupofallelementsfixedbyθ.
Moreover,assumethattheflip-flopsystemCθisconnected.ThenGθactstransitively
onCθifandonlyifthereexistsachamberc∈CθsuchthatforeachpanelPinCθ
containingcthestabilizerStabGθ(P)actstransitivelyonP.

5.3.Transitivityinrank1
Bytheprecedingsection,itisnaturaltostudytransitivitypropertiesofrank1
groups,i.e.,ofMoufangsets.Forthis,webuildontheworkdoneinChapter3.
Specifically,foranontrivialinvolutoryautomorphismθofaMoufangset(briefly:
aflip),twoquestionsareofinteresttous:

(1)WhendoesGθacttransitivelyontheflip-flopsystemCθ?Aswearelooking
atrank1(i.e.,aMoufangset),Cθconsistsofallpointsnotfixedbyθ.
(2)Whendoestheflip-flopsystemequalthewholeMoufangset?Thisisofhigh
relevancewhenstudyingIwasawadecompositions.

InSection5.3.1,wefocusonprojectivelinesoverfields,thesimplestkindofMo-
ufangsets,associatedtothegroupPSL2resp.SL2.Whilebeingfarfromgeneral,
thisalreadysufficestodealwithflipsofalllocallysplithigher-rankgroups,inpartic-
ular,splitalgebraicandKac-Moodygroups.Wewillsoonseethatflipsofprojective
linescorrespondcloselytosesquilinearforms,andtheflip-flopsystemthentothe
anisotropicpointsofthisform.Thissimpleinsightisthekeytoouranalysis.
Wealsopresentsomelimitedresultsonflipsofprojectivelinesoverskewfields,
usingMoufangsettechniques,inSection5.3.2.Thismightserveasencouragement
forfutureworkinthisdirection.
TheresultspresentedinthissectionarebasedonjointworkwithTomDeMedts
andRalfGramlichin[DMGH09],buthavebeenextended.

91

5.Transitiveactionsonflip-flopsystems
5.3.1.Transitivityinrank1:SL2andPSL2
Inthissection,wecloselyfollowthenotationfromSection3.1.Inparticular,Fis
afield,GisSL2(F)orPSL2(F),andθisanarbitrarynontrivialinvolutoryauto-
set,morphismthepro(i.e.,jectivaeflip)lineofP1(GF.)=∼HenceG/Bθ+.alsoAgain,inducesweaflipdenoteofbytheVasso=F2ciatedtheMoufastandardng
moduleofG.
Inthissetting,weobtainthefollowingdescriptionoftheflip-flopsystemassociated
toθ:Weknowthatitconsistsofthesetof1-dimensionalvectorsubspacesofV
(correspondingtopointsofP1(F))whicharemovedbyθ.InSection3.1,wesaw
thethateverysesquilinearflipisformconjugatefδ,σt(andoaitsstandardassociatedflipθσδ,σ,-quadraticwhichintformurnqδis,σ).closelyForGrelated=SL2to,
thealgebraicversionofthisconnectionisthatthefixedpointgroupoftheflip
correspondstothesubgroupofelementspreservingtheform.Geometrically,theflip
θisthepolarityinducedbytheformfδ,σ.Accordingly,thefixedpointsarethose
1elemen-dimensionaltsofthesubspacesflip-flopwhicsystemharearetheisotropicanisotropicwith1respectto-dimensionalthisthissubspaces.formwhilethe
ThisobservationtogetherwiththerestofSection3.1allowsustocharacterize
whenθδ,σisflip-floptransitive,i.e.,whenitsfixedpointgroupactstransitivelyon
subspaces.-dimensional1anisotropictheRemark5.3.1.Inthefollowing,allresultsarewrittenwithG=PSL2(F)inmind.
ThecorrespondingresultsforG=SL2(F)canbeobtainedbyreplacingPKδ,σby
Kδ,σandrestrictingεto+1inbothstatementsandproofs.
Definition5.3.2.LetFbeafieldwithanautomorphismσoforderatmost2.We
definethenormmapNσasfollows:Nσ:F→FixF(σ):a→aaσ.
Thusfor(ba)∈Vwehaveqδ,σ((ba))=Nσ(b)−δNσ(a).
Lemma5.3.3.Aflip-floptransitiveflipofGisisconjugatetoθ−ε,σwithε∈
{+1,−1}.
Prflopoof.systemByPropcorrespositiononds3.1.4,totheit1suffices-dimensionaltodealwithsubspacesflipsofwhicthehareformθδ,σanisotropic.Thewithflip-
respecttoqδ,σ.Clearly(10)and(01)areanisotropic.0x
forBysomethex∈assumedF∗.SincetransitivitPKδ,σyofPpreservKδ,σes,thethereformexistsqδ,σg∈upPtoKδa,σsignsuchε∈that{g+1(,1−)1=},(0w)e
evha1=qδ,σ((10))=εqδ,σ(g(10))=εqδ,σ((0x))=−εδNσ(x),
thusδ=−εNσ(x−1).LetX:=(01x0),andsetIntX(g):=XgX−1,theinnerauto-
morphisminducedbyX.ThenIntX◦θδ,σ◦IntX−1equalsθ−ε,σ.
Bytheprecedinglemma,itsufficestodeterminewhenθ1,σandθ−1,σareflip-flop
e.transitiv92

5.3.Transitivityinrank1

Proposition5.3.4.Theflipθ−1,σ(resp.θ+1,σ)isflip-floptransitiveifandonlyif
thesum(resp.difference)ofanytwonormsisεtimesanorm,forε∈{+1,−1}.
Proof.Wepresenttheargumentforδ=−1;thecaseδ=+1workscompletely
.analogouslyAssumeθ−1,σisflip-floptransitive.Takeanarbitraryanisotropicvector(ab)∈F2.
Bytransitivity,thereexistsg∈PK−1,σsuchthatg(ba)=(x0)forsomex∈F∗.
SincePKδ,σpreservestheformquptoasignε∈{+1,−1},wehave
Nσ(a)+Nσ(b)=−δNσ(a)+Nσ(b)=qδ,σ((ba))=εqδ,σ(g(ba))=εqδ,σ((x0))=εNσ(x),
provingthatasumoftwonormsisεtimesanorm.
Conversely,supposethatthesumoftwoarbitrarynormsisknownatobeεtimes
anorm.Itsufficestoshowthatforanyanisotropicvectorv=(b)∈F2there
existsg∈PK1,σmappingvtoanonzeromultipleof(10).Choosexsuchthat
εNσ(x)=Nσ(a)+Nσ(b)=qδ,σ(v)forε∈{+1,−1},andnotethatx=0sincevis
anisotropic.Thustheequation
b−a
−εxaσεbxσba=x0
xxfinishestheproof,asthematrixonthelefthandsideoftheequationisinPK−1,σ.
Example5.3.5.ConsiderG=PSL2(R).Thenbothθ−1,idandθ+1,idareflip-flop
transitive:Inthefirstcase,wehavetoverifythatthesumoftwosquaresisagain
asquare,whichiscertainlytrue(sothisflipisinfactalsoflip-floptransitiveover
SL2(R)).Inthesecondcase,wereadilyseethatthedifferenceoftwosquaresisalways
eitherasquareorminusasquare,hencebythetheprecedingproposition,thisflipis
alsoflip-floptransitive(butonlyoverPSL2(F),notoverSL2(R)).However,thetwo
flipsarecertainlynotconjugate,becauseforthefirstone,allpointsareanisotropic
(andtheflip-flopsystemequalstheprojectiveline)whileinthesecondcase(11)
and(−11)areisotropicpoints.
Inthespecialcasethat−1isanorm,wecanrefinethisrequirementabit.The
proofforthefollowingPropositionwaspartiallyinspiredby[AG06,Lemma4.2].
RecallfromProposition3.1.5,iftwostandard−1flipsθδ,σandθε,τareconjugate,then
thereexistsρ∈Aut(F)suchthatσ=ρτρ.Animmediateconsequenceisthat
Nσ(F)=(Nτ(F))ρ.Therefore,theproperty“−1isanorm”iswell-definedforan
arbitraryflipofSL2orPSL2.
Proposition5.3.6.Consideraflipθconjugatetosomestandardinvolutionθδ,σ,
andsuppose−1isanormwithrespecttoθ.Thenθisconjugatetoθ−1,σ.Moreover,
θisflip-floptransitiveifandonlyifthenormmapNσissurjective(ontoFixF(σ)),
orσ=idandcharF=2.

93

5.Transitiveactionsonflip-flopsystems

Proof.ByPropositions3.1.4and5.3.4,ourflipisconjugatetoθ−1,σorθ+1,σ.Since
−1isanorm,thesetwoareactuallyconjugateaswell.
Supposeourflipisflip-floptransitive.ByProposition5.3.4andsince−1isanorm,
sumsanddifferencesofnormsareagainnorms.Moreover,productsandquotients
of(nonzero)normsarenorms.WeconcludethatthenormsNσ(F)formasubfield
ofFixF(σ).
Pickx∈Fsuchthatxσ=−x(thisisalwayspossibleunlessσ=idandcharF=
2).Sincethenormsformasubfield,foranyy∈FixF(σ)wehavethat
Nσ(xy+1)−Nσ(xy)−Nσ(1)=xy+(xy)σ=(x+xσ)y
isagainanorm.Since(x+xσ)=0,weconcludethatFixF(σ)=Nσ(F).
Conversely,ifσ=idandcharF=2,thenthenorms(i.e.,squares)formasubfield
ofF(assquaringisafieldendomorphismincharacteristic2).Inparticular,their
sumsareagainnorms.Ontheotherhand,ifthenormfunctionissurjective,we
haveFixF(σ)=Nσ(F),andthusthesumofnormsisanorm.Ineithercase,by
Proposition5.3.4,θ−1,δisflip-floptransitive.
Remark5.3.7.Ifσ=idandcharF=2,thenthenormmapissurjectiveifand
onlyifFisquadraticallyclosed.
Forfinitefieldswithnontrivialfieldautomorphismoforder2itiswell-knownthat
theassociatednormmapisalwayssurjective.

Wenowturntothesecondquestion:Whenisaflipbuildingtransitive,i.e.,when
doesitscentralizeracttransitivelyonthewholeprojectiveline?Thisisequivalent
toaskingwhentheassociatedformisanisotropic,whichimmediatelyrulesoutall
finitefieldsFqwithq≡3mod4(becausethen,−1isasquare).Tobeprecise,we
wing:follotheobtainProposition5.3.8.Aflipistransitiveonthewholeprojectivelineifandonlyifit
isconjugatetoθ−1,σ,thesumoftwonormsisεtimesanormforε∈{+1,−1},and
−1isnotanorm.
Proof.Letθbeaflip.Wehavetoverifytwothings:Theflip-flopsystemmustequal
thewholeprojectiveline,andtheflipmustbeflip-floptransitive.
ByLemma5.3.3transitivityimpliesthatourflipisconjugatetoeitherθ−1,σor
θ+1,σ.Forthelatter,(11)isisotropicwhile(01)isnot,hencewecannothavea
transitiveactiononthewholeprojectivelinepreservingtheform.Soθmustbe
conjugatetoθ−1,σ.
If−1wasanorm,say−1=Nσ(x),thenthevector(x1)wouldbeisotropic,and
theflip-flopsystemwouldnotequalthewholeprojectiveline.Conversely,ifthereis
anonzeroisotropicvector(ba),then0=Nσ(a)+Nσ(b),andw.l.o.g.b=0,hence
Nσ(ba)=−1.
TheclaimnowfollowsfromProposition5.3.4.

94

5.3.Transitivityinrank1

5.3.2.Transitivityinrank1:Moufangflips
ThissectionbuildsontheworkdoneinSection3.2andthebasicsetuppresented
inSection1.10,anddealswiththetransitivityofarestrictedclassofflipsofthe
MoufangsetassociatedtoPSL2overaskewfield.Itisbasedon[DMGH09,Section
5].Theresultshavebeenobtainedindependentlybytheauthorofthepresentthesis
usingmatrixcomputations:Essentially,onecanextendtheworkdoneintheprevious
sectionandinSection3.1toskewfields,butthecomputationsbecomealotmore
involvedwithoutprovidingamajornewinsight.However,weprefertoincludethis
Moufangsetbasedapproachasitmaybemoreconceptualthantheonebasedon
matrixcomputations.Additionally,itillustrateshowMoufangsettheorycanhelp
problem.thissolvinginDefinition5.3.9.Ifτ2=id,thenϕ=1isaflipautomorphism.Wewillcallthe
correspondingautomorphismθ1ofG(asdefinedinSection3.2)theobviousflip.
Observethatθ1isjustconjugationbyτ.
Definition5.3.10.Aflipautomorphismϕ∈Aut(U)iscalledfullytransitiveif
thegroupGθϕistransitiveonX.
LetM=M(U,τ)beaMoufangsetwithτ2=id.Thentheobviousflipθ1isfully
transitiveifandonlyifCG(τ)istransitiveonXbecauseGθ1=CG(τ).
Lemma5.3.11.LetM=M(U,τ)beaMoufangsetwithτ2=id,andassumethat
theobviousflipisfullytransitive.Thenτhasnofixedpoints.
Proof.Assumethataτ=aforsomea∈U∗.Letg∈CG(τ)besuchthat0g=a.
Then∞g=0τg=0gτ=aτ=a=0gandhence∞=0,acontradiction.
WenowexaminethetransitivityoftheobviousflipforM(D)whereDisanarbi-
field.ewsktraryDefinition5.3.12.Ifg=(cadb)∈GL2(D),thentheDieudonnédeterminantdet(g)∈
D∗/[D∗,D∗]isdefinedas
det(g):=ab:=ad−aca−1bifa=0;
cd−cbifa=0;
see[Die43].ThenSL2(D)ispreciselythekerneloftheDieudonnédeterminant,i.e.,
amatrixg∈GL2(D)liesinSL2(D)ifandonlyifdet(g)∈[D∗,D∗].Alsoobserve
thatdet(λg)≡det(gλ)≡λ2det(g)mod[D∗,D∗]forallλ∈D∗.
Lemma5.3.13.LetG=SL2(D)andletτ=(−0101)∈G.Then
C(τ)=aba2+aba−1b∈[D∗,D∗]ifa=0;
G−bab2∈[D∗,D∗]ifa=0
PCG(τ)=ab∙(2a2+∗aba−1∗b)∈[D∗,D∗]ifa=0where=±1.
−ba∙b∈[D,D]ifa=0

95

5.Transitiveactionsonflip-flopsystems

Proof.Thisisastraightforwardcalculation.
Proposition5.3.14.LetG=SL2(D)andletτ=(−0101)∈G.LetXbethe
projectivelineoverD,i.e.,X={(ba)D=0|a,b∈D}.Thenthefollowingare
quivalent:e(1)CG(τ)istransitiveonX;
(2)a2+aba−1b∈(D∗)2[D∗,D∗]foralla,b∈D∗;
(3)1+a2∈(D∗)2[D∗,D∗]foralla∈D∗.
Proof.Sincea2+aba−1b=a2(1+a−1ba−1b),wehavea2+aba−1b∈(D∗)2[D∗,D∗]if
andonlyif1+a−1ba−1b∈(D∗)2[D∗,D∗].Equivalencebetween(ii)and(iii)follows
byreplacinga−1bbyainthelatterterm.
Assumenowthat(ii)holds.Leta,b∈D∗bearbitrary;wewanttoshowthat
thereexistssomeg∈CG(τ)mapping(ba)to(0z)forsomez∈D∗.By(ii),we
knowthatthereissomec∈D∗suchthatb−2+b−1a−1ba−1≡c−2mod[D∗,D∗].Let
g:=−cbca−−11cbca−−11.Thendet(g)≡c2(b−2+b−1a−1ba−1)≡1,i.e.,g∈G.Moreover,
g(ba)=(0z)forz=c(b−1a+a−1b),provingthatCG(τ)actstransitivelyonX.
Conversely,assumethatCG(τ)actstransitivelyonX.Leta,b∈D∗bearbitrary;
thenthereexistssomeg∈CG(τ)mapping(01)Dto(ba)D,i.e.,thereissomez∈D∗
suchthatgmaps(0z)to(ba).ByLemma5.3.13,weknowthatghastheform
g=(−xyxy)withx2+xyx−1y∈[D∗,D∗].Theng(0z)=(−xzyz),andhencea=xzand
b=−yz.Hencea2+aba−1b=xzxz+xzyx−1yz=xzx−1∙(x2+xyx−1y)∙z,and
sincex2+xyx−1y∈[D∗,D∗],thisimpliesa2+aba−1b≡xzx−1z≡z2mod[D∗,D∗].
Sincea,b∈D∗werearbitrary,thisproves(ii).
Proposition5.3.15.LetG=PSL2(D),letτ=(−0110)∈SL2(D),andaletτ˜bethe
imageofτinG.LetXbetheprojectivelineoverD,i.e.,X={(b)D|a,b∈
D,notbothzero}.Thenthefollowingareequivalent:
(1)CG(τ˜)istransitiveonX;
(2)PCG(τ)istransitiveonX;
(3)a2+aba−1b∈{±1}∙(D∗)2[D∗,D∗]foralla,b∈D∗;
(4)1+a2∈{±1}∙(D∗)2[D∗,D∗]foralla∈D∗.
Proof.Theequivalencebetween(i)and(ii)followsimmediatelyfromthedefinition
oftheprojectivecentralizerPCG(τ).Theotherequivalencesareshownexactlyasin
theproofofProposition5.3.14above.
Corollary5.3.16.(i)LetG=SL2(D),andassumethatforalla∈D∗,wehave
1+ha∈H.ThenCG(τ)actstransitivelyonX.
(ii)LetG=PSL2(D),andassumethatforalla∈D∗wehave1+ha∈{±1}∙H.
ThenCG(τ˜)actstransitivelyonX.

96

5.4.Iwasawadecompositions

Proof.Weonlyshow(i).Theproofof(ii)iscompletelysimilar.Soleta∈D∗
bearbitrary,andassumethat1+ha=h∈H.Then1+1ha=1h,i.e.,1+
a2=1h.Writeh=hx1∙∙∙hxnwithx1,...,xn∈D∗.Then1h=xn∙∙∙x1∙1∙
x1∙∙∙xn≡(x1∙∙∙xn)2mod[D∗,D∗],andhence1+a2=1h∈(D∗)2[D∗,D∗].So
(iii)ofProposition5.3.14holds,andthereforethegroupCG(τ)actstransitivelyon
.XAnaturalextensionofthestudyoftheobviousflipwouldbetostudyitsclose
relatives,thesemi-obviousflips,whichareobtainedbycomposingtheobviousflip
withanautomorphismoranti-automorphismofD.

5.4.Iwasawadecompositions
TheworkinthissectionisbasedonjointworkTomDeMedtsandRalfGramlichin
[DMGH09].TheconnectedIwasawasemisimpledecompsplitositionrealofLieagroupconnectedisoneofsemisimplethemostcomplexfundamenLietalgroupobservora-a
tionscomplexofresp.classicalsplitLierealtheoryLie.ItgroupGimpliesisconthattrolledthebygeometryanyofmaximalaconnectedcompactsemisimplesubgroup
orK.theExamplestransitiveareWactioneyl’sofKonunitarianthetricTitskinbuildingtheG/Brepresen.IntationthecasetheoryofoftheLieconnectedgroups,
estingsemisimpleepimorphismsplitrealfromLiethegrouprealoftypebuildingG2ofthetyplatteerG2,theimpliessplittheCayleyexistenceohexagon,faninonter-to
SOthe4(Rreal)→buildingSO3(Rof),typcf.eA2,[Gra98].therealThisproejectivpimorphismeplane,bycannotmeansbeofdescribtheedusingepimorphismthe
groupoftypeG2becauseitisquasisimple.
Tobeabletotransfertheseideastoabroaderclassofgroups,weextendthenotion
ofanIwasawadecompositioninthefollowingway:
Definition5.4.1.AgroupGwithatwinBN-pair(B+,B−,N)admitsanIwasawa
decompositionifthereexistsaproperbuilding-transitivequasi-flipθofG.
whicInhothmapserwsomeords,pGositivadmitseBoranelIwgroupasawatoandecompoppositeositionone,ifandtheresuchexiststhatθ∈Amoreout(vGer)
G=GθB+whereGθisthecentralizerofθinG.
OurinterestinIwasawadecompositionsstemsfromthepresentationbygenera-
torsandrelations(inthenon-finitelypresentedcaseusuallyformulatedasauniversal
enmainveloonpingsomeresultsimplyofanconnectedamalgam)ofsimplicialanarbitrarycomplex,groupwhichactingiswithimpliedabyfundamenTits’talLemmado-
of[Paas85,complexLemmaLie5],group[Tit86,orCorollarycomplex1].Kac-MoTheodytransitivgroupeactionontheofaassocompactciatedrealcomplexform
buildingcompactgivrealesformparticularlyistheunivniceersalpresenenvtationselopingasgroupstudiedofinthe[GGH]amalgamand[Gra06];consistingtheof

97

5.Transitiveactionsonflip-flopsystems

therank1and2subgroupswithrespecttoasystemoffundamentalroots.The
followingtheoremisthemainamalgamationresultofthepresentwork.
Theorem5.4.2(jointworkwithGramlichandDeMedts,see[DMGH09]).Consider
acenteredRGD-system(G,{Uα}α∈Φ,T)withaninvolutionθsuchthatG=GθBis
anIwasawadecompositionofG(cf.Definition5.4.1).Furthermore,letΠbeasystem
offundamentalrootsofΦandfor{α,β}⊆ΠletXα,β:=Uα,U−α,Uβ,U−β.
ThenθinducesaninvolutiononeachXα,βandGθistheuniversalenveloping
groupoftheamalgam((Xα,β)θ){α,β}⊆ΠoffixedpointsubgroupsofthegroupsXα,β.
Proof.ByLemma5.1.13theinvolutionθinducesaninvolutionofeachgroupXα,β.
BytheIwasawadecompositionthegroupGθactswithafundamentaldomainonthe
simplicialcomplexΔassociatedtoG/B,theflagcomplexofG/B.ChooseFtobe
afundamentaldomainofΔstabilizedbythetorusTofG,sothatthestabilizersof
thesimplicesofFofdimension0andonewithrespecttothenaturalactionofG
onΔareexactlythegroups(Xα)θTand(Xαβ)θT.Bythesimpleconnectednessof
buildinggeometriesofrankatleast3(cf.[Bro89,TheoremIV.5.2]or[Tit74,Theorem
13.32])andTits’Lemma(seee.g.[Pas85,Lemma5],[Tit86,Corollary1])thegroup
Gθequalstheuniversalenvelopinggroupoftheamalgam((Xαβ)θT)α,β∈Π.Finally,
by[GLS95,Lemma29.3]thetorusTcanbereconstructedfromtherank2toriTαβ,
α,β∈Π,andsothegroupGactuallyequalstheuniversalenvelopinggroupofthe
amalgam((Xαβ)θ)α,β∈Π.
Iwasawadecompositionshavebeenstudiedforallkindsofgroups(cf.[Bel],[Krö])
andoverrealclosedfields(cf.[Gro72]).Inthissectionwecharacterizethefields
FforwhichagroupwithanF-locallysplitrootgroupdatumadmitsanIwasawa
decomposition,cf.Definition5.4.1andTheorem5.4.7.Wepointoutthatthisclass
ofgroupscontainstheclassofgroupsofF-rationalpointsofaconnectedreductive
algebraicgroupdefinedoverF(cf.[Spr98])aswellastheclassofsplitKac-Moody
groupsoverF(cf.[Rém02],[Tit87]).
ForthenextdefinitionrecallthatanyCartan–Chevalleyinvolutionof(P)SL2(F)
isgiven,resp.inducedbythetranspose-inverseautomorphismwithrespecttothe
choiceofabasisofthenaturalSL2(F)-moduleF2.
Definition5.4.3.LetFbeafield,letσbeanautomorphismofFoforderatmost
2,let(G,{Uα}α∈Φ,T)beanF-locallysplitRGD-system.Wecallanautomorphism
θofGaσ-twistedChevalleyinvolutionofGifitsatisfiesforallα∈Φ:
(1)θ2=idG,
(2)Uαθ=U−α,and
(3)θ◦σinducesthestandardChevalleyinvolution(resp.itsimageunderthe
canonicalprojection)onXα:=Uα,U−α=∼(P)SL2(F).

98

5.4.Iwasawadecompositions

AllsplitKac-Moodygroupsadmitσ-twistedChevalleyinvolutions(inparticular,
theclassicalChevalleyinvolutionanditstwistunderaninvolutoryfieldautomor-
phism)bycombiningasignautomorphismwithafieldautomorphism,see[CM05,
Section8.2].Likewiseforallsplitreductivealgebraicgroups.Alsogroupswith2-
sphericalF-locallysplitrootgroupdatumoverafieldwithatleastfourelements
condition:thismeetLemma5.4.4.LetFbeafieldwithatleastfourelements,letσbeanautomorphism
ofFoforderatmost2,let(G,{Uα}α∈Φ,T)beacentered,2-sphericalandF-locally
splitRGD-system.ThenGadmitsaσ-twistedChevalleyinvolution.
Proof.By[AM97](andalsobytheunpublishedmanuscript[Müh96])thegroupGisa
universalenvelopinggroupoftheamalgamα,β∈ΠXα,βforasystemΠoffundamental
rootsofΦ,sothatanyautomorphismofα,β∈ΠXα,βinducesanautomorphismofG.
Foreachpairα,β∈ΠtheChevalleyinvolutionofthesplitreductivealgebraicgroup
Xα,βcomposedwithσinducesautomorphismsθαonXαandθβonXβsatisfyingthe
criteriaforaσ-twistedChevalleyinvolution.Thereforethereexistsaninvolution
oftheamalgamα,β∈ΠXα,βinducingθαonXαforeachα∈Φ.Consequently
thereexistsaninvolutionθonitsuniversalenvelopinggroupGinducingθαon
eachsubgroupXα.ThisinvolutionθofGbyconstructionisaσ-twistedChevalley
involutionofG.
Whatmakesσ-twistedChevalleyinvolutionsinterestingisthattheyareflips.In
particulartheycentralizetheWeylgroup.Hencewecanapplyourfullmachinery
them.toProposition5.4.5.Anyσ-twistedChevalleyinvolutionθofagroupGisaBN-flip.
Proof.Bydefinition,θisaninvolution.Furthermore,theBorelsubgroupB+is
generatedbyTandthesetofrootgroupsassociatedtothepositiverootsystem
Φ+⊂Φ.Moreprecisely,B=T.Uα|α∈Φ+.SinceT=α∈ΦNG(Uα)by[CR08,
Corollary5.3],theinvolutionθstabilizesTandmapsB+toB−=T.U−α|α∈Φ+.
Finally,θactstriviallyonW=N/TaseachrootαoftherootlatticeofWismapped
ontoitsnegative−α,whichmeansthatthereflectiongivenbyαismappedontothe
reflectiongivenby−α,whichisidenticaltothereflectiongivenbyα.
ThefollowingcorollaryisadirectconsequenceofProposition5.3.8,onceapplied
toSL2(byrestrictingεto1),andoncetoPSL2.
Corollary5.4.6(jointworkwithDeMedtsandGramlich).ThegroupPSL2(F)resp.
SL2(F)admitsanIwasawadecompositionifandonlyifFadmitsanautomorphism
σoforderatmost2suchthat
(1)−1isnotanorm,and
(2)asumofnormsisanorm(intheSL2(F)case),resp.asumofnormsisε
timesanorm,whereε∈{+1,−1}(inthePSL2(F)case),

99

5.Transitiveactionsonflip-flopsystems

withrespecttothenormmapNσ:F→FixF(σ):x→xxσ.
Wefinallyhaveassembledalltoolsrequiredtoproveourmainresultinthissection.
Theorem5.4.7(jointworkwithGramlichandDeMedts).LetFbeafieldandlet
(G,{Uα}α∈Φ,T)beanF-locallysplitRGD-system.ThegroupGadmitsanIwasawa
decompositionifandonlyifFadmitsanautomorphismσoforderatmost2such
that(1)−1isnotanorm,and
(2)(i)ifthereexistsarank1subgroupUα,U−αofGisomorphictoSL2(F),
thenasumofnormsisanorm,or
(ii)ifeachrank1subgroupUα,U−αofGisisomorphictoPSL2(F),thena
sumofnormsis±1timesanorm,
withrespecttothenormmapNσ:F→FixF(σ):x→xxσ,and
(3)Gadmitsaσ-twistedChevalleyinvolution.
Proof.AssumetheexistenceofanIwasawadecompositionofG.Bydefinitionthere
existsaninvolutionθofGsuchthatG=GθB+.HenceanyBorelsubgroupofG
ismappedontoanoppositeone,sothatbyLemma5.1.13theinvolutionθcen-
tralizestheWeylgroupN/Tand,foranysimplerootα,normalizesthegroup
Xα:=Uα,U−α,whichbyF-localsplitnessisisomorphicto(P)SL2(F).Inpar-
ticulartherestrictionθ|XαofθtoXαisaBN-flip.
WenowarguethatthisrestrictedBN-flipinducesanIwasawadecompositionof
Xα.LetPαbethepanelofthebuildingcorrespondingtotherootα.ByCorollary
5.2.3weknowthat(GPα)θ=GPα∩GθactstransitivelyonPα,anditremainsto
showthatthisisalsothecasefor(Xα)θ=Xα∩Gθ.FirstobservethatP−α=θ(Pα)
andhence(GPα)θalsostabilizesthepanelP−α.For,ifg∈(GPα)θ,theng.P−α=
g.θ(Pα)=θ(g.Pα)=θ(Pα)=P−αandsog∈(GPα)θ=GPα∩GP−α∩Gθ.If
x∈(GPα)θstabilizesthechamberB+inPα,thenx.B−=x.θ(B+)=θ(x.B+)=
θ(B+)=B−.Weconcludethatx∈B+∩B−=T.Moreover,thegroupUα<Xα
stabilizesB+andactstransitivelyonP−α.Thus,infact(GPα)θ=(XαT)∩Gθ.Any
t∈T\XαactstriviallyonPα.Hence,since(GPα)θactstransitivelyonPα,sodoes
(Xα)θ.AccordinglyXαadmitsanIwasawadecomposition.
Therefore,byCorollary5.4.6below,thefieldFadmitsanautomorphismσwith
erties.proprequiredtheFortheconverseimplication,letθbetheσ-twistedChevalleyinvolutionofG.
Foreachα∈ΦtheinvolutionθinducesaBN-flipθαonXα.ByProposition
5.3.8below,theseinducedflipsaretransitive.HencebyCorollary5.2.3,wehave
G=GθB+,provingthatGadmitsanIwasawadecomposition.
Corollaries6.1.6and6.2.4specializethistheoremtothecaseofalgebraicand
groups.dyoKac-Mo

100

5.4.Iwasawadecompositions

Remark5.4.8.Allsplitrank2groupsareknown.Thisfollowsfromtheclassifi-
cationofMoufangpolygons(see[TW02]andalsotheenumerationinSection2.6),
butalsomoreelementarybypre-classificationresults(e.g.byresultsonCheval-
leygroups,see[Ste68b]).Inparticular,theirrank1groupsarenotisomorphicto
PSL2(F),exceptforPSL2(F)×PSL2(F)orPSL2(F)×SL2(F).
Thus,ifallrank1groupsareisomorphictoPSL2(F),thenwecandeducethat
thediagramofthegroupmustberightangled,i.e.,anytwonodesareeithernot
joinedbyanedge,orbyanedgewithinfinityaslabel.Suchexamplescanbe
obtainedbytakingarbitrarydirectproductsofPSL2(F)withitself,orusingcertain
freeconstructions(seee.g.[CR08,Example2.8]orformoredetails,[RR06]).
Inviewoftheaboveremark,weobtainthefollowingcorollary:
Corollary5.4.9.LetFbeafield,let(G,{Uα}α∈Φ,T)beanF-locallysplit2-spherical
RGD-systemwithoutisolatednodesinthediagram.ThegroupGadmitsanIwasawa
decompositionifandonlyifFadmitsanautomorphismσoforderatmost2such
that(1)−1isnotanorm(inparticular,charF=2)and,
(2)asumofnormsisanorm,
withrespecttothenormmapNσ:F→FixF(σ):x→xxσ,and
(3)Gadmitsaσ-twistedChevalleyinvolution.

5.4.1.FieldsadmittingIwasawadecompositions
BesidesthewidelyknownIwasawadecompositionsoverrealclosedfields(see[Gro72])
andthefieldofcomplexnumbersthereexistlotsoffieldsadmittingautomorphisms
thatsatisfytheconditionsofCorollary5.4.6.Notethatanypythagoreanformally
realfieldFsatisfiesthe√conditionsofCorollary5.4.6withrespecttotheidentity
automorphismasdoesF[−1]withrespecttothenontrivialGaloisautomorphism.
InthePSL2(F)casethefinitefieldsFqwithq≡3mod4yieldadditionalexamples.
Quiteanumberofpropertiesofpythagoreanandformallyrealfieldsareknown,
j93].[Ra[Lam05],[Lam73],seeRemark5.4.10.(1)AfieldisformallyrealpythagoreanifandonlyifitsWitt
groupistorsionfree,see[Lam05,TheoremVIII.4.1].
(2)Afieldisformallyrealpythagoreanifandonlyifitistheintersectionofa
nonemptyfamilyofeuclideansubfieldsofitsalgebraicclosure,see[Lam05,
I.4.4].IVITheorem(3)IfafieldFisformallyrealpythagorean,thensoisthefieldF((t))offormal
Laurentseries,see[Raj93,Theorem18.9].

101

5.Transitiveactionsonflip-flopsystems

(4)Ifna+1fieldFisrealclosed,thenthefieldF((t1))∙∙∙((tn))ispythagoreanandhas
2squareclasses,see[Raj93,Theorem18.9].
(5)IfFispythagoreanbutnotformallyreal,thenFisquadraticallyclosed,see
[Rathej93,fieldofTheoremthenumb16.4].ersInwhichareparticular,theconstructibleintersectionwithofstraighthetedgerealnandumbcoersmpasswith,
ispythagoreanandformallyreal.
(6)IfFformallyisafieldreal)ifinandwhichonly−1ifisFnotdoesanotsquare,admitthenanyitiscyclicpexythagoreantensionof(andorderhence4,
[DD65].seeInspiredbyclassicalLietheoryandthepassagefromcomplexLiegroupstotheir
splitrealforms,thequestionariseswhetheranIwasawadecompositionG=GθBof
θainvgroupolvingGawithnonantrivialF-lofieldcallyasplitroutomorphismotgroupσ:Fdatum→Fwalwithaysrespectimpliestoantheinvexistenceolution
inofvanolvingIwasathewatrivialdecompfieldositionovautomorphismerthefieldonFixFixFF((σσ)).withTherespfollowectingtoanexampleinvshoolutionws
thatthisisgenerallynotthecase.
fourExamplesquare5.4.11.classes.LetSucFbheafieldsformallyexist,realseefieldforwhicexamplehisnotp[Szy75].ythagorThiseanandmeansadmitsthat
aexactlyuniquetwoordering.squareChoclassesoseaconptainositiveabsolutelynon-squapreositivelemenetelemenw∈Fts,.soSetthatα:=there√−wexistsand
F˜:=F[α].Then
N(x0+αx1)+N(y0+αy1)=x02+wx12+y02+wy12,
whicHencehisthereaexistnon-negativz0andezn1uminbFer,suchhencethateitherasquareorasquaremultipleofw.
N(x0+αx1)+N(y0+αy1)=x02+wx12+y02+wy12=z02+wz12=N(z0+αz1)
andconditionsthustheoffieldCorollaryF˜together5.4.6whilewithFthenontogethertrivialwithGaloistheidentityautomorphismdoesnot.satisfiesthe

5.5.Moreonflipsoflocallysplitgroups
Forstrongflipsoflocallysplitgroups,flip-floptransitivityimpliesdistancetransi-
.ytivitsitiveLemmaquasi-flip5.5.1ofa(GramliclocalhlyandsplitRMühlherr)GD-system.Supp(oseG,{θUαis}αa∈Φstr,T)ong,thenandθisflip-flopdistanctran-e
ansitive.tr

102

5.5.Moreonflipsoflocallysplitgroups

Proof.LetcanddbechambersofC+withidenticalθ-codistancew.Ifw=1W
thenwearetransitiveonCwθbyhypothesis.Assumenowbymeansofinduction
thattransitivityhasalreadybeenestablishedforeachθ-codistanceofshorterlength.
Thereexistss∈Ssuchthatl(sw)<l(w).ByLemma1.3.2eithersw=wθ(s)or
swθ(s)isaθ-twistedinvolutionofshorterlengththanw;denoteitbyw.Accordingly
byLemma2.4.2thereisachambera∼scwithδθ(a)=wandalsoachamberb∼sd
withδθ(b)=w.Byinductionthereexistsanelementg∈Γmappingatob.Thus
bothdandg(c)arecontainedinthesames-panelPs.
ViewingPsasthegeometryof1-dimensionalsubspacesofa2-dimensionalvector
spaceendowedwithanontrivialσ-sesquilinearformf,bothdandg(c)correspond
tof-singular1-dimensionalsubspaces.ByWitt’sTheoremthereexistsanelement
h∈Γmappingg(c)ontod.
Remark5.5.2.Thehypothesisoftheprecedinglemmacanbesomewhatweakened
byreplacing“strong”withtherequirementthatuniformdescentispossible,asgiven
inthefollowingdefinition.ByLemma2.4.2everystrongquasi-flipallowsuniform
t.descenDefinition5.5.3.Letθbeaquasi-flip.Forachamberc,wedefinethedescent
setDθ(c):={s∈S|∃d∈Ps(c):lθ(d)<lθ(c)}.Wesayθallowsuniform
descentifforanytwochambersc,dwithequalθ-codistancethesetsDθ(c)and
Dθ(d)coincide.

103

5.

eransitivT

104

actions

on

flip-flop

systems

CHAPTERSIX

APPLICATIONSTOALGEBRAICANDKAC-MOODY
OUPSGR

Throughoutthiswholechapter,wewillalwaysassumeallfieldstobeofcharacteristic
differentfrom2unlessstateddifferently.

groupsbraicAlge6.1.Inthissectionwepresentapplicationsoftheresultsintheprecedingchaptersof
thepresentworktoreductivelinearalgebraicgroups.Foranintroductiontolinear
algebraicgroups,wereferto[Hum75],[Bor91],[Spr98].
LetGbeaconnectedreductivelinearalgebraicgroupdefinedoveraninfinitefield
F.WedenotethesetofF-rationalpointsofGbyG(F).Inparticular,weidentify
GwithG(F),whereFisthealgebraicclosureofF.
AssumethatGisisotropicoverF,i.e.,somepropernontrivialparabolicsubgroup
ofGisdefinedoverF.LetTbeamaximalF-splitF-torus.ByBorelandTits
[BT65],thereexistsafamilyofrootgroups{Uα}α∈Φ,indexedbytherelativeroot
systemΦof(G(F),T(F)),suchthat(G(F),{Uα}α∈Φ,T(F))isanRGD-system.For
detailsseee.g.[BT72,Section6],[AB08,Section7.9].
Inparticular,Gadmitsa(twin)BN-pair.Ingeneral,thegroupB(whichwe
somewhatmisleadinglyhavealsocalled“Borelgroup”inpreviouschapters)willbe
thegroupP(F)ofF-rationalpointsofaminimalparabolicF-subgroupPofG.
TherootgroupsofG(F)areuniquely2-divisibleifandonlyifcharF=2.Indeed,
iftheF-rankofGisatleast2,thisfollowsfromProposition2.6.1.Butingeneral
thisistruebecausetherootgroupsarevectorspacesoverForextensionsofavector
spaceoverFbyanothersuchvectorspace(seee.g.[BT73,Section8]).Yetanother
argumentcanbeseeninthefollowingsketchedproof.
Lemma6.1.1.LetUα(F)bearootgroupofaconnectedreductivegroupdefinedover
afieldF.ThenUα(F)isuniquely2-divisibleifandonlyifcharF=2.

105

6.ApplicationstoalgebraicandKac-Moodygroups

Sketchofproof.NotethatUα(F)isconnected(seee.g.[Hum75,26.3]).Takean
grouparbitrarygeneratedelemenbtyuu∈Uinsideα(F)G,(Fand)inletVtersected:=uwith∩GG((FF)).beSincetheU(ZariskiF)isclosureclosed,Vofisthea
subgroupofitandhenceofUα(F).IfcharF=0,thenVisaαconnected1-dimensional
unipotentgroupisomorphictotheadditivegroupofF.IfcharF=p>2,then2Vis
afinitecyclicp-group.Ineithercasethereexistsauniquev∈Vsuchthatv=u.
Uniquenessthenfollowsbytheobservationthatu=v.

6.1.1.Quasi-flipsofalgebraicgroups
Inthesisthistosectionarbitraryweinshovwolutthatorywecanautomorphismsapplytheoffullconnectedmachineryreductivdeveelopgroupsedinorthisof
finitegroupsofLietype.Readerswhoareonlyinterestedintheconsequencesfor
algebraicgroupswhichwecandrawfromthismaywishtoskiptothenextsection.
positivLeteFFbe-rankan(i.e.,infinitefieldisotropic).andItGisaadeepsimpleresultalgebraicbyBorelgroupandTitsdefinedo[BT73]verF(seeandalsoof
of[Ste73]thegroforupanG(F)EnglishofFsummary-rationalpofointhetsscenplittralsintoaresults)productthatofaabstractfieldautautomorphismsomorphism,
anF-isogenyandtheinverseofapurelyinseparablecentralisogeny(thelattercan
beomittedifGissimplyconnectedoradjoint).
FAsa-subgroupstoconsequence,suchangroupsyofabstractthesametypautomorphisme.Infact,ofG(thisF)evmapsenextends(minimal)tosemisparabimpleolic
groups(andfromtheretoreductivegroups):
algebrPropaicositiongroup6.1.2overan(Propinfiniteosition7.2fieldinF.[BT73])Then.LanyetGabstrbeaactconnectedautomorphismreductiveofGline(Far)
parmapsabolicparFab-subolicgrFoups-subgrareoupsmappagainedtotoppararababolicolicFF-sub-subgrgroups).oups(inparticular,minimal
Overfinitefields,reductivealgebraicgroupsbecomesplit,sothatthereexist
BorelF-subgroups.However,theconnectedcomponentbecomestrivial,sostudying
ofG(FLie)istypenot(seethee.g.right[Car72]),approach.whichThearecorrectobtainedviewpfromointisalgebraictostudygroupsfiniteviagrLang’soups
theorem.LetGnowbeafinitegroupofLietypecomingfromareductivealgebraic
F-group.Letp:=charF.By[Che55]oneknowsthattheBorelsubgroupscanbe
abstractlyautomorphismdescribwilledasmaptheBorelnormalizersgroupstoofpBorel-Sylowgroupssubgroups,(seealsoand[Ste60hence]).anyAllinabstractall,
get:ewFact6.1.3.LetFbeaninfinitefieldandGthegroupofF-rationalpointsofasemi-
anysimpleabstralgebractaicgrautomorphismoupdefineofdGoverofForderand2ofispaositivequasi-flipF-rankas(i.e.,definedisotrinopic).ChapterThen2.
ThesameholdsifGisafinitegroupofLietype.

106

ebraicAlg6.1.groups

6.1.2.Applicationstoalgebraicgroups
Here,wegeneralize[HW93,Proposition6.10]inthesensethatwealsocovernon-
linearautomorphisms.Proposition6.8inloc.cit.canbegeneralizedanalogously.
Corollary6.1.4(ofCorollary2.7.3andProposition2.6.1).SupposeGisaconnected
isotropicreductivealgebraicgroupdefinedoverafieldFwithcharF=2,andPa
minimalparabolicF-subgroup.LetθbeanabstractinvolutoryautomorphismofG.
Let{Ai|i∈I}berepresentativesoftheGθ(F)-conjugacyclassesofθ-stablemaximal
F-splittoriinG.Then
Gθ(F)\G(F)/P(F)=∼WGθ(F)(Ai)\WG(F)(Ai).
I∈i

TheInworkChapterdone5wetherestudiedappliesIwasasawfolloawsdecomptosplitositionsalgeofbraicgroupsgroups:withatwinBN-pair.
anIwDefinitionasawa6.1.5decomp(Cf.5osition.4.1).ifAthereductreivexistseanalgebraicabstractgroupinvGolutorydefinedoverautomorphismFadmitsθ
ofandG(GF()F)and=Gaθ(F)Pminimal(F)whereparabGolicθ(FF)is-subgroupthecenPoftralizerGofsucθhinthatG(Fθ().P)isoppositeP
Corollary[DMGH09]).L6.1.6etF(ofbeaTheoremfieldand5.4.7;letGbjoinetawsplitorkcwithonnectedGramlicrehductiveandDealgebraicMedts,groupsee
tiondefineGd(F)over=GF.θ(FThe)B(grF)oupifofandF-ronlyationalifFpadmitsointsGan(F)admitsautomorphismanσIwasawaoforderdecatompmostosi-
thatsuch2(1)−1isnotanorm,and
(2)(i)eitherasumofnormsisanorm,or
(ii)asumofnormsisεtimesanorm,whereε∈{+1,−1},(andthiscase
canonlyoccurifallrank1subgroupsofGareisomorphictoPSL2(F)),
withrespecttothenormmapNσ:F→FixF(σ):x→xxσ.
Remark6.1.7.InSection5.4.1,wegaveexamplesandsomeextradetailsonfields
(2)(ii)satisfyingbecomesthevcriteriaacuousgivifenthereabovaree.noNoteisolatedthatnoindesviewinofthediaRemarkgramof5.4.8,G(F).condition
LetGbeaconnectedreductiveF-group,andletθbeanabstractinvolutory
automorphismofG(F).In[HW93]itisshownthatallminimalθ-splitparabolic
F-subgroupshaveequaltype,providedθisanalgebraicmorphism(seePropositions
4.8and4.11inloc.cit.).
Ppreciselyartof,itourwimpliesorkinthefolloChapterwing4can(recallbethatFconsideredistoassumedbeatovbeariationinfinite):ofthis.More

107

6.ApplicationstoalgebraicandKac-Moodygroups

Propinvolutoryosition6.1.8.automorphismLetGofbeGa(Fc).onneIfctethedrediagrductiveamofFG-gr(Foup,)candontainsletθnobeantripleabstrbondsact
(i.e.,therearenoresiduesisomorphictoG2),thenallminimalθ-splitparabolic
F-subgroupshaveequaltype.
Proresiduesof.ofMinimaltheθsame-splittypeparab(cf.olicFSection-subgroups4.3).Bycorrouresphypondotheseone-to-onesandtoTheoremminimal4.1.10,Phan
allminimalPhanresidueshaveequaltype.

flipsinearL6.1.3.Definition6.1.9.LetGbethegroupofF-rationalpointsofaconnectedreductive
F-group.AabstractinvolutoryautomorphismθofGwhichisan(algebraic)F-
morphismiscalledF-linearquasi-flip.
InvolutoryF-morphismsarewell-understoodobjects,andtheirtheoryheavilyin-
fluencedusduringthecreationofthepresentthesis,inparticular[HW93].Assuch,
wedonotsaymuchaboutthesehere,butratherjustgiveonebriefexample.
ThefollowingobservationisduetoHelminckandWang,asaconsequenceof
Propositions4.8and4.11of[HW93].(SeealsoProposition3.16of[Hel97]and
[Spr86].)RecallthatGisidentifiedwiththerationalpointsoverF,thealgebraic
.FofclosureFact6.1.10.LetGbeaconnectedreductivelineargroup,letθbeanF-linearquasi-
flipofG.Thenallminimalθ-splitparabolicF-subgroupsofGareconjugateunder
.GθGeometrically,thismeansthatGθactstransitivelyonthesetofminimalPhan
residuesofthebuildingG/B(whereBisaBorelsubgroupofG).Inthespecialcase
thattheminimalθ-splitparabolicF-subgroupsareBorelsubgroups,thismeansthat
Gθactstransitivelyontheflip-flopsystem.

flipsmi-linearSe6.1.4.Definition6.1.11.LetGbeareductivelinearF-group.Asemi-linearquasi-flip
ofGisaninvolutoryF-automorphismofGcomposedwithafieldautomorphismof
.FNotethatthefieldautomorphismnecessarilyhasorderatmost2.
Lemma6.1.12.Semi-linearquasi-flipsofsplitreductivegroupsarestrong,i.e.,
possesstheDevillers-Mühlherrproperty.
Proof.Informally,theargumentisthatinasplitgroup,projectingfromapanelto
anotherpanelis“linear”,andthereforecomposingthiswithasemi-linearmapcan
notbetheidentity.

108

6.2.GroupsofKac-Moodytype

Lessinformally,letGbeasplitreductivegroupandθasemi-linearflipofG.
proAssumejP(θ(cP))is=ac.IfpanelPisinnotC+.parallWeelhatoveitstoshoimagewθ(Pthat)(i.e.,theretheexistsprocjection∈Psucfromhθ(Pthat)
toPisasinglechamber),thenthisisclearlythecase.SowemayassumethatP
proandθ(jectionP)areonθpa(c),rallel.i.e.,WthatewillθdonoeswnotassumepossessthattheθmapsallDevillers-MühlherrchambersinpropPerttoy,theandir
willleadthistoacontradiction.
canLetchocboseeaacθham-stablebertinwinPandapartmendenotetΣitsconθ-cotainingdistancecandbyw.henceByθ(c).Theorem1Denote2.5.8thewe
rootuniqueincΣhamconbertaininginΣc∩Pbutnotdifferenc.tByfromourcbycassumptionandletthatα=θ(doα+es,α−not)bpeosthesesstthewin
Devillers-Mühlherrpropertywehaveδθ(c)=w=δθ(c).
ΣWatecoclaimdistancethatwα−∩fromθ(Pc)=resp.{θc(c.)}.TheIndeedreflectionθ(c)sαresp.θcorresp(c)isondingthetouniqueαincterchamhaberngesin
cByandconcv,exitythereforeoftwinitmroustotswalsoeswdeduceapθ(cthat)andθ(cθ)(∈c)α,,soasθ(itPis)cisinloserthetobcthanoundaryθ(cof)αis..
spondingConsequentoα.tlyByθ(αrig+)idit=yαof−tandwinthusbuildingsθ(α)θ=(Uα.)=LetUUαb=eUthe.rootgroupcorre-
αθ(α)α
ofdLettodθ∈(PP)\by{cd}..FByoranouryu∈standingUαwehavassumption,eu.d∈dP=\{θ(cd})..DeSincenoteantheyproelementjectionof
Gpreservesdistancesandcodistances,inparticularumustpreserveprojections.
P\{Thereforec},u.daccordingly=projPu(=u.dθ()u=)θand(u.dU)=⊂θG(u.).d.ButUαactssharplytransitivelyon
θαSinceautomorphismGissplit,σofUαF.isDenoteisomorphicU:=to(θthe◦σ)(Uadditiv).eBygrouprigiditofFy.ofNowθsphericalinvolvesabuildings,field
αbUe=theUβinvforersesomeoftherootβlinear.Butmap(thenθ◦σthe):Unon-linear→U,mapwhichσ:isUβabsurd.→Uαwouldhaveto
βαgroupRemarkandtheir6.1.13.Thesemi-linearproofwejustautomorphisms.gaveappliesThealatterlsomaappliesybetodefinedsplitKac-Mocompletelyody
lemofanalogouslysplitasKac-Moforodyalgebraicgroups,groups,seethanks[Cap05],tothe[CM05],solution[CM06].oftheisomorphismprob-

6.2.GroupsofKac-Moodytype
Inthefollowing,wegiveonlyaveryroughoverviewonKac-Moodygroups.The
interestedreadermayfindagentleintroductionin[AB08,Section8.11],or[CR08,
Section3.3].TheoriginalreferencesforthekindofKac-Moodygroupsweconsider
areofcourse[Tit87]and[Tit92].
1WhileTheorem2.5.8requirescharacteristicdifferentfrom2,onecanshowthatsuchatwin
theapartmenexptosition,alwawysejustexistsapifptheealtoourquasi-flipisstandingseconmi-linear.ventionHowandevaser,sumeforthethecsakeofharacteristicsimplicittoybofe
.2fromtdifferen

109

6.ApplicationstoalgebraicandKac-Moodygroups

Chevalley’sworkmadeitpossibletodefinegroupsoverarbitraryfieldsanalogously
tothecomplexsemisimpleLiegroupsbydefiningthesegroupsintermsofagroup
functororgroupscheme.Parallelingthisgroundbreakingwork,Titsintroducedin
[Tit87]asimilardescriptionofKac-Moodygroups.Roughlyspeaking,heintroduced
functorgroupaGB:RING→GROUP
fromthecategoryofcommutativeringswithunitintothecategoryofgroupsde-
pendingonaintegralfiniterootbasisB.Thisgroupfunctorhasthepropertythat
forafieldF,thegroupGB(F)possessesanaturalrootgroupdatum,withrootgroups
isomorphictotheadditivegroupofF.ThisfunctoriscalledtheTitsfunctor.IfF
isafield,thegroupGB(F)iscalledsplitKac-MoodygroupoverF.
Beingequippedwitharootgroupdatum,anyKac-Moodygroupalsopossessesa
twinBN-pair.However,unlikealgebraicgroups,thetwogroupsB+andB−arein
generalnotconjugateanymore;rather,theyareconjugateifandonlyiftheWeyl
groupoftheRGD-systemisfinite.
Recently,CapraceandMühlherrhavedeterminedallabstractautomorphismsof
(infinite)splitKac-Moodygroupsoveralmostarbitrarygroundfields(onlyF2and
F3requiresomeextracare),see[Cap05],[CM05],[CM06].Theirresultessentially
states(intheirreduciblecase)thatanysuchautomorphismsplitsintotheproduct
ofaninnerautomorphism,asignautomorphism(theidentitymaportheChevalley
involution,whichinterchangestheconjugacyclassesofpositiveandnegativeBorel
groups),adiagonalautomorphism,agraphautomorphism(generalizingdiagram
automorphisms),andafieldautomorphism.Inthegeneralcase,thestatementis
morecomplicated.Nevertheless,thefollowingholds:
Fact6.2.1.LetGbeaninfiniteKac-MoodygroupoversomefieldF,|F|≥4.
ThenanyabstractautomorphismofGmapsBorelsubgroupstoBorelsubgroups.In
particular,anyinvolutoryautomorphismofGeitherpreservesthesignofallBorel
subgroups,oritinterchangesplusandminustypeBorelsubgroups.
Thisfollowsfrom[Cap05,Theorem4.1],whichimpliesthatanygroupautomor-
phismofGinducesanautomorphismoftherootgroupdatumofG.
Wecannowgeneralize[KW92,Proposition5.15](whichinloc.cit.wasonlystated
forKac-Moodygroupsoveralgebraicallyclosedfieldsincharacteristic0)asfollows:
Corollary6.2.2(ofCorollary2.7.3,Proposition2.6.1andExample2.5.5).Suppose
GisasplitKac-MoodygroupdefinedoverafieldFwithcharF=2.Letθbe
aninvolutoryautomorphismofGinterchangingthetwoconjugacyclassesofBorel
groupsofG.ThenwiththenotationfromProposition2.7.2,wehave
Gθ\G/B=∼WGθ(Ai)\WG(Ai).
I∈iInChapter5westudiedIwasawadecompositionsofgroupswithatwinBN-pair.
TheworkdonethereappliesasfollowstosplitKac-Moodygroups:

110

6.2.GroupsofKac-Moodytype

Definition6.2.3(Cf.5.4.1).AKac-MoodygroupGdefinedFadmitsanIwasawa
decompositionifthereexistsanabstractinvolutoryautomorphismθofGanda
cenBoreltralizersubgroupofθinBGof.Gsuchthatθ(B)isoppositeBandG=GθBwhereGθisthe
Corollary[DMGH09]).L6.2.4etF(ofbeafieldTheoremand5.4.7;letGjoinbetawsplitorkKwithac-MooGramlicdygrhoupandoverDeF.Medts,ThenseeG
σofadmitsorderanatIwasawamost2decsuchompthatositionG=GθBifandonlyifFadmitsanautomorphism
(1)−1isnotanorm,and
(2)(i)eitherasumofnormsisanorm,or
(ii)caansumonlyofoccurnormsifisallεranktimes1asubgrnorm,oupsofwherGeεare∈{+1,isomorphic−1},to(andPSLthis2(Fc)),ase
withrespecttothenormmapNσ:F→FixF(σ):x→xxσ.
IncriteriaSectiongivenab5.4.1,ove.wegaveexamplesandsomeextradetailsonfieldssatisfyingthe

6.2.1.LocallyfiniteKac-Moodygroups
Inthissection,wecollectafewresultsaboutlocallyfiniteKac-Moodygroups.
Thefollowingisknownforcertainspecialcases.E.g.whenθissemi-linearand
thediagramissphericalthisfollowsfromLang’stheorem(andGθthenisinfacta
finitegroup),evenforoddq.SeealsoRemark6.2.8below.
Theorem6.2.5.SupposeGisasplitKac-Moodygroupoftype(W,S)overafinite
fieldFq,q≥5andodd,with2-sphericaldiagram(andnoG2residue).Letθbea
quasi-flipofG,i.e.,aninvolutoryautomorphismofGwhichinterchangesthetwo
conjugacyclassesofBorelgroups.ThenthecentralizerGθofθinGisfinitely
d.ategenerProof.Foranyc∈C,thestabilizerStabGθ(c)alsostabilizesθ(c),henceiscontained
inStabG(c)∩StabG(θ(c)).Thusitisaboundedsubgroup(i.e.,theintersectionof
twosphericalparabolicsubgroupsofoppositesign)asdefinedin[CM06].SinceGis
locallyfinite,Corollary3.8inloc.cit.impliesthatStabGθ(c)isfinite.Bythesame
argument,thetorusT=B+∩B−isfinite.Sinceqisodd,Theorem2.5.8ensures
thatallchambersarecontainedinaθ-stableapartment.ThusbyLemma5.1.12,Gθ
actswithfinitelymanyorbitsO1,...,OnonCθ.
Chooseachamberc1∈O1.Foreachi∈{2,...,n}pickachamberci∈Oisuch
thatl(c1,ci)isminimalamongallchambersinOi.Setm:=1+maxi∈{2,...,n}l(c1,ci),
andletXbethesetofallchambersatdistanceatmostmfromc1.ClearlyX
containsalltheciandalltheirpanels.Sinceourbuildingislocallyfinite,thisisa
finiteset.Byconstruction,XintersectsallGθorbits.

111

6.ApplicationstoalgebraicandKac-Moodygroups

LetY:={g∈Gθ|g.X∩X=∅}.SinceXisfiniteandallchamberstabilizers
inGθarefinite,Yisalsoafiniteset.LetH:=YandθconsiderthesetHθ.X.
Thisisreadilyseentobeconnected.Assumetherewasc∈C\H.X.SinceCis
connectedbyTheorem4.1.10,wecanchooseaminimalgalleryinsideCθfromcto
somechamberinH.X.Byfollowingthisgallery,wefindachambercoutsideH.X
butadjacenttoachamberdinsideH.X.
ButbydefinitionofHandX,theremustbesomeh∈Handsomeorbitrep-
resentativecisuchthatd=h.ci.Butthend∈h.X,andbyconstructionalsoall
panelsofdarecontainedinh.X,thusinparticularc∈h.X.Contradiction,hence
Gθ.X=H.X.SincemoreoverHcontainsStabGθ(ci)forallorbitrepresentativesci,
weconcludethatGθ=H=Yisfinitelygenerated.
Remark6.2.6.TheexclusionofG2residuesisadeficiencyofTheorem4.1.10,which
hopefullycanberemovedinthefuture.
Ontheotherhand,therestrictiontocharacteristic2ispartiallyduetoTheorem
2.5.8whichweusetoconstructθ-stableapartmentsaroundarbitrarychambers.
Thereisnohopeofimprovingthisboundusingourmethodsaslongasθisnot
furtherrestricted.Howeverifθisassumedtobesemi-linearthentheexistenceof
θ-stableapartmentsisactuallyguaranteedregardlessofthecharacteristic.Thusin
thiscaseitsufficestoknowthatCθisconnected(asasubstituteforTheorem4.1.10)
toconcludethatGθisfinitelygenerated.
LatticesWewilldemonstratethatTheorem6.2.5isinsomesensesharp(withthepreceding
remarkinmind)bysketchingthatforKac-Moodygroupswhicharenot2-spherical
thegroupGθwillingeneralnotbefinitelygenerated.Forthiswefirstneedanother
resultshowingthatGθisalatticeinG+.Recallthatalatticeisadiscretesubgroup
ΓofalocallycompactgroupGwiththepropertythatΓ\Gisendowedwitha
finiteG-invariantmeasure.MoreoverG+denotesthetopologicalcompletionofGas
definedin[CR09,Section1.2],whichthereisshowntobealocallycompactgroup.
ThetheoremwestatenowisaslightlymodifiedversionofaresultbyGramlich
andMühlherr[GM08].Wemerelyextendtheclassofmorphismstowhichitapplies.
Inloc.cit.theresultisgivenonlyfordistancetransitiveflipswiththeDevillers-
Mühlherrproperty.WecanreplacethisassumptionbyLemma5.1.12(combined
withTheorem2.5.8)atthepriceofhavingtorestricttocharacteristicdifferentfrom
2.Moreover,GθwillisadiscretesubgroupbecauseGθ∩U+=Gθ∩U+∩U−is
aboundedsubgroupofG,hencefinite.Withthisinminditisstraightforwardto
adjusttheproofgiveninloc.cit.totheversionofthetheoremwepresenthere.
Theorem6.2.7(GramlichandMühlherr,2007).SupposeGisasplitKac-Moody
groupoftype(W,S)overafinitefieldFq,q≥5andodd.Letθbeaquasi-flipofG,
i.e.,aninvolutoryautomorphismofGwhichinterchangesthetwoconjugacyclasses
seriesw∈Wql(w)converges.
ofBorelgroups.1ThenthecentralizerGθofθinGisalatticeinthegroupG+ifthe

112

6.2.GroupsofKac-Moodytype

LetGbealocallyfiniteKac-Moodygroupoftype(W,S).ThenGisalways
finitelygenerated.ForGθtobefinitelygenerated,wehadtoassumethatGis2-
spherical.InfactCaprace,Gramlich,andMühlherrhaverecentlyobservedthatGθ
maynotbefinitelygeneratedifGisnot2-spherical:LetTbeatreeresidueofthe
building,thenG.Tisasimplicialtreeby[DJ02,Proposition2.1].Thekeyinsightis
thefollowing:TheactionofthelatticeGθonthesimplicialtreeG.Tisminimalbut
thereareinfinitelymanyGθ-orbitsonG.TifInvθ(T)={δθ(c)∈W|c∈T}isinfinite
(whichforinstanceisthecaseifθisasemi-linearflip).From[Bas93,Proposition
7.9](also[BL01,Proposition5.6])itfollowsthatthelatticeGθcannotbefinitely
generated.IfGis2-spherical,thenitisfinitelypresented.ForGθtobefinitelypresented,
ingeneralweneedGtobeatleast3-spherical.2This“gap”betweenGandGθis
believedtoextendtohigherfinitenessproperties.
Remark6.2.8.By[DJ02]inthe2-sphericalcasethefullautomorphismgroupof
thebuildingassociatedtolocallyfiniteKac-MoodygroupGhasKazhdan’sproperty
(T)providedthegroundfieldissufficientlylarge(e.g.ifitsorderisgreaterthan
1764n/25,wherenisthedimensionofthebuilding).SinceGθisalatticeinGitalso
hasproperty(T)by[BdlHV08,Theorem1.7.1]andinparticularisfinitelygenerated
by[BdlHV08,Theorem1.3.1].Notethattheboundsinloc.cit.areknowntobenot
optimal.

2Hereisabriefargumentforthis,atleastintheaffinecase:By[GM08]thegroupGθisa
lattice.Henceby[Mar91,ChapterIX]itisS-arithmeticintheambientsemisimpleLiegroup
(thecompletionoftheaffineKac-Moodygroup).Now[BW07]statesthatanS-arithmetic
subgroupofasplitsemisimplealgebraicgroupoverafunctionfieldisoftypeF2ifandonly
ifthecorrespondingaffinediagramis3-spherical.So,ifthediagramis2-spherical,butnot
3-spherical,thenthegroupisnotfinitelypresented.

113

6.

ationsApplic

114

to

algebraic

and

Kac-Mo

o

dy

oupsgr

APPENDIXA

TSRESULCOMPUTER

Inthisappendix,wepresentresultsobtainedwiththehelpofmachinecomputations,
aswellasthecomputercodethatwasused.Allcomputationswereperformedwith
[GAP08].GAPofhelpthe

A.1.ConnectednessofRθ:θ-acutequadrangles
InthissectionwestudythegeometryoppositeaMoufangsubsetofapanelinseveral
low-ordergeneralizedquadrangles,namelythoseoforder(3,9),(4,8)and(4,16).To
dothis,wefirstproduceacomputerrepresentationofthepointsandlines,byloading
thefilequadrangle.gapintoGAP,withthevariables(n,q)setto(4,3),(5,2)and
(4,4),respectively.
Itiseasytocomputethegeometryoppositeachamber,apointoralinewith
getsthiscosmallerde.Iftwheefixmoreappanel,ointssawyethehavpeointotberowoppofaositeline,of.thenHencethisitoppsufficesositetotakgeometryeall
maximalMoufangsubsetsofthepanel,thenshowthatforeachofthemtheopposite
geometryisconnected.Thecodeinquadrangle-acute.gapdoesjustthat,printing
outany“counterexamples”itfinds.Byrunningitweconfirmedthatthesubsetsof
thequadranglesdescribedabovearealwaysconnected.

A.2.ConnectednessofRθ:θ-parallelprojectiveplanes
InProposition4.6.11,westudiedconnectednessoftheflip-flopsystemassociatedto
polaritiesofprojectiveplanes.Onecasewasleftopen,namelythepolaritiesofthe
projectiveplaneoverthefieldwiththreeelements.Whilethisisatinyexampleand
certainlycouldbehandledbymanualcomputations,wepresentsomecomputercode
dealingwiththisproblemforthisandotherprojectiveplanes.
Infiletriangle.gap,wecomputetheprojectiveplaneT(q)overthefieldFq,
whereqisaprimepowersetbytheuser.Then,infiletriangle-invs.gap,alltype

115

resultsterCompuA.

OrderDescriptionMaxlθ|P||L||C|#comps
2Prolinearjectiveplaneautomorphism-374472181
linearpolarity33361
3Projectiveplane-131352
linearlinearpolarityautomorphism339898242411
4Projectiveplane-2121105
linearlinearpolarityautomorphism3315161516606411
semilinearsemilinearpolarityautomorphism3312141214245641
5Projectiveplane-3131186
linearlinearpolarityautomorphism332524252411202011
7Projectiveplane-5757456
linearlinearpolarityautomorphism334948494833363611
8Projectiveplane-7373657
linearautomorphism364645121
linearpolarity363635041
9Projectiveplane-9191910
linearautomorphism380807201
linearpolarity381817201
semilinearautomorphism378787021
semilinearpolarity363633781
TableA.1.:Sizesoftheflip-flopsystemsinvariousfiniteprojectiveplanes.

preservingautomorphismsofthetrianglearedetermined,aswellasanorthogonal
polarity.ThissufficestocomputethefullextendedautomorphismgroupofT(q)
aandinrepresentheretativalleiscconjugacyhosenandclassestheofinflip-flopvolutions.systemisFinally,computed.foreachInTconjugacyableA.1,class,we
presentthecomputedresultsforseveralsmallprojectiveplanes.Notethatitconfirms
thattheonlyexceptionforconnectednessofRθoccursintheprojectiveplaneoforder
4withasemilinearprojectivity.

A.3.ConnectednessofRθ:θ-parallelquadrangles
Inthissection,wecompletetheproofofProposition4.6.18.Forthis,westudy
involutoryautomorphismsofcertainlowordergeneralizedquadrangles.Specifically,
foreachconjugacyclassofinvolutions,wepickarepresentativeθanddetermine
whetherRθ(thesetofchambersmovedmaximallybyθ)isconnectedasachamber
system.Thequadranglesweneedtoconsiderareofthefollowingorders:(s,s)for
s∈{2,3,4,5,7,8,9,16};(s2,s)fors∈{2,3,4};and(s2,s3)fors∈{2,3,4}.
Thecodeworksasfollows:Inthefilesquadrangle.gapandquadrangle2.gap,
thereiscodewhichcomputesinternalrepresentationsofquadranglesoforthogonal
andunitarytype.Theninquadrangle-invs.gap,representativesfortheconjugacy

116

A.3.ConnectednessofRθ:θ-parallelquadrangles

(s,s)DescriptionMaxlθ|P||L||C|#comps(s2,s)DescriptionMaxlθ|P||L||C|#comps
(2,2)linearQuadrangle3-121515824451(4,2)linearQuadrangle3-32452427135961
linear3812241semilinear33012601
linear488162semilinear42412481
(3,3)Quadrangle-4040160linear41616324
linearlinear33243032401209611(9,3)linearQuadrangle3-252280112112100811201
linear43624961semilinear3240727201
linear41824483semilinear4270727201
(4,4)Quadrangle-8585425linearsemilinear44144216967257657611
linearlinear338064648032032011linear4180603601
linear464642561(16,4)Quadrangle-11053255525
semilinear440408010linear3102432051201
(5,5)Quadrangle-156156936semilinear3102024040801
linear31301567801linearsemilinear447689602562403830724011
linear31201447201
linearlinear4411005012012048072011(25,5)linearQuadrangle3-3327615075675618900196561
semilinear33120600156001
(7,7)linearQuadrangle3-340050400400280032001semilinear43250600156001
linear333638426881linearsemilinear4423000400720600144001440011
linearlinear442394923363362016268811linear42600520104001
(8,8)Quadrangle-5855855265
linear357651246081(s2,s3)DescriptionMaxlθ|P||L||C|#comps
linear351257646081
linear451251240961(4,8)linearQuadrangle3-128165288297115214851
(9,9)Quadrangle-8208208200semilinear41201807201
linear373882073801linear4642565121
linear372080072001
linearlinear446848107207205760720011(9,27)linearQuadrangle3-224401606720683266832004801
semilinear454054032401linearsemilinear441216051258326048351840628811
(11,11)Quadrangle-1464146417568
linear313421464161041(16,64)linearQuadrangle3-16384174256656066625106496011326251
linearlinear431452132013201440115840584011semilinear416320612009792001
linear412101320132001linear412288655367864321
(13,13)Quadrangle-2380238033320
linear322102380309401
linear321842352305761
linearlinear442028236621842184320576620811
(16,16)linearQuadrangle3-435243694096436967427396321
linear340964352696321
linearsemilinear443264409632409664639165536811

TableA.2.:Sizesoftheflip-flopsystemsinvariousfinitequadrangles.

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resultsterCompuA.

classesoftype-preservinginvolutionsarechosen.Thelatterismostlydonebybrute
force,insteadofrelyingonclassificationresultsfortheseinvolutions,astosimplify
theorderco(sde2,sand3)rineduceevenctheriskharacteristic,forerrors.whereAnweuseexceptionresultsismadefromfor[YP92]theandquadrangles[AS76]ofin
ordertobeabletocomputeallinvolutionsforthequadrangleoforder(16,64).
SeeTableA.2forasummaryoftheoutputoftheGAPcode.Foreachofthe
linesquadranglesandchamabboveers)(andofthesomequadrangle.more),itgivMoreoesvtheer,sizeforeac(i.e.,htheclassnofumbinverofolutorypoinau-ts,
tomorphismsofthesequadrangles,itstateswhetheritislinearorsemilinear,and
numpresenbertsofthepoints,maximallinesnandumericalchambθers)-distanceoftheaccorresphamberondingismovflip-floped,thesystemsizeR(i.e.,θ,andthe
itsnumberofconnectedcomponents.Itisevidentthatthereisonlyoneconnected
componentinRθ,exceptforthequadranglesoforder(2,2),(3,3),(4,4)and(4,2),
claimed.as

coGAPA.4.de

triangle.gap##q:=3;Projectiveplaneoforderq
Fn:=:=3;GF(q);
VG:=:=GL(n,q);FullRowSpace(F,n);
s:=MAX_DISTq;:=3;
q;:=tPrint("Projectiveplaneoforder",q,"...");
#pointsCompute:=allOrbit(G,points,[[1,0,0]]lines*andOne(F),chambersofOnSubspathetrianglecesByCanonicalBasis);;
Print(Size(points),"points,");
linesPrint(Size(:=Orbit(G,lines),"li[[1,0,0],nes,");[0,1,0]]*One(F),OnSubspacesByCanonicalBasis);;
n.gap");Read("commoAssert(0,Size(points)=1+q+q^2);
Assert(0,Assert(0,Size(chSize(lines)ambers)==1+q+q(1^2);+q)*(1+q+q^2));

quadrangle.gap#Quadranglesoforder(q^2,q)and(q^2,q^3)
#q:=2;4;:=#n5;:=#nGF(q^2);:=FGU(n,q);:=GFullRowSpace(F,n);:=V4;:=MAX_DIST

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q^2;:=st:=q^(1+(n-4)*2);#qifn=4,andq^3ifn=5
Print("Quadrangleoforder(",s,",",t,")...");
x#:=ElementFirst(F,xsuchg->thatg^(q+1)x^(q+1)==-One-1(F));
then4=niflinepoint:=:=[[1,x,0,0],[[1,x,0,0]]*[0,0,1,x^-One(F);1]]*One(F);
elselinepoint:=:=[[1,x,0,0,0],[[1,x,0,0,0]]*[0,0,0,1One(F);,x^-1]]*One(F);
fi;#pointsCompute:=allSet(Orbit(G,ppoints,linesandoint,OnSubspacesByCachambersofthenonicalBasis));;quadrangle
linesPrint(Size(:=points),Set(Orbit(G,li"points,");ne,OnSubspacesByCanonicalBasis));;
Print(Size(lines),"lines,");
n.gap");Read("commoAssert(0,Size(points)=(1+s)*(1+s*t));
Assert(0,Assert(0,Size(chSize(lines)ambers)==(1+t)(1*(1+s*t));+s)*(1+t)*(1+s*t));

decoGAPA.4.

quadrangle2.gap(q,q)orderofQuadrangles##q:=2;Fn:=:=5;GF(q);
GV:=:=GO(n,q);FullRowSpace(F,n);
4;:=MAX_DISTts:=:=q;q;
Print("Quadrangleoforder(",s,",",t,")...");
#bmatCompute:=theInvariantBilineproductoftwoarForm(G).matrix;vectorsw.r.t.theformusedbyGAP
Prod:=function(u,v)returnu*bmat*TransposedMat(v);end;;
#Computetheproductoftwovectorsw.r.t.theformusedbyGAP
qmatQForm:=:=function(u)InvariantQuadrareturnu*qmat*ticForm(G).matrix;TransposedMat(u);end;;
#Computeallpoints,linesandchambersofthequadrangle
One(F);*[[0,1,0,0,0]]:=pointpointsPrint(Size(:=points),Set(Filtered("points,");Orbit(SL(n,q),point,OnSubspacesByCanonicalBasis),p->QForm(p)=[[Zero(F)]]));;
linepoint2:=:=[point[1],First(points,point2[1]]p->Is;Zero(p[1][1])andIsZero(p[1][2])andProd(point,p)=[[Zero(F)]]);;
linesPrint(Size(:=lines),Set(Orbit(G,li"lines,");ne,OnSubspacesByCanonicalBasis));;
n.gap");Read("commoAssert(0,Assert(0,Size(liSize(pones)ints)==(1+t)(1+s*(1+s*t));)*(1+s*t));

119

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terCompuA.results

Assert(0,Size(chambers)=(1+s)*(1+t)*(1+s*t));

common.gapchambersallCompute#chambers:=List(Set(Union(NormedRowVecList([1..Size(lines)]tors(VectorSpace(F,,l->lines[l])),
)));;v->[Position(points,[v]),l])
chambers\n");"(chambers),Print(Size#pencilDetermine:=allList(points,pointrowsp->[])and;;pencils.Usedtospeedupcomputationslateron.
forpointrowcin:=chambersList(lines,dol->[]);;
AddSet(pointAddSet(pencil[c[1]],row[c[2]],c[2]);c[1]);
od;

quadrangle-acute.gapn:=4;q:=3;#n:=5;q:=2;#n:=4;q:=4;angle.gap");Read("quadr#Lookbothatapencil,andapointrow.
panels:=[List(pencil[1],l->[1,l]),List(pointrow[1],p->[p,1])];
dopanelsinpanelfor#Sizeofmoufangsubsetminusonemustdividesizeofthissetminusone.
#Usethistodeterminethemaximalsize’k’aMoufangsubsetcouldpossiblyhave.
l)-1)[1]);FactorsInt(Size(pane1+(Size(panel)-1)/(:=ksubsets:=Filtered(Combinations(panel,k),comb->IsSubset(comb,panel{[1,2]}));;
tmp:=Filtered(chambers,c->IsOpposite(panel[1],c)andIsOpposite(panel[2],c));;
forTinsubsetsdo
TOp:=Filtered(tmp,c->ForAll(T,cT->IsOpposite(cT,c)));;
ifNrComponentsChamberSet(TOp)<>1thenDisplay(T);fi;
od;od;

vs.gaptriangle-inRead("poly-utils.gap");#Somehelpercoder
group:automorphismfullCompute###1)DetermineapermutationpresentationofPGU(n,q)onourpoints
phi:=ActionHomomorphism(G,points,OnSubspacesByCanonicalBasis);
#2)Computeactionoffrobeniusautomorphism
gfrob:=Permutation(FrobeniusAutomorphism(F),points,OnTuplesTuples);
#3)Thefullautomorphismgroup:
Image(phi);:=GpermautG:=Group(Concatenation(GeneratorsOfGroup(Gperm),[gfrob]));
##Computetheextendedautomorphismgroup
##Functionwhichcomputestheorthogonalcomplementofavector
#w.r.t.theformgivenbytheidentitymatrix.
on(v)functi:=orPolarOfVectmat;localmat:=MutableCopyMat(BaseMat([[v[2],-v[1],0],[v[3],0,-v[1]],[0,v[3],-v[2]]]*One(F)));

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at(mat);TriangulizeMmat;returnend;;#Computethepolarsofallthechambers,for*some*polarity.Fromthis,
#wecanget*all*polarities,sincetheproductoftwopolaritiesisa
#typepreservingautomorphism,soinautG.
p2l:=List(points,p->Position(lines,PolarOfVector(p[1])));;
l2p:=List([1..Size(lines)],l->Position(p2l,l));;
tau:=PermList(List(chambers,c->Position(chambers,[l2p[c[2]],p2l[c[1]]])));
groupautomorphismExtended#psi:=ActionHomomorphism(autG,chambers,OnChambers);
Assert(0,autGOnChambersautGOnC:=Image(psi);hambers^tau=autGOnChambers);
extG:=Group(Concatenation(GeneratorsOfGroup(autGOnChambers),[tau]));
#Determinerepresentativesforallinvolutions
Print("Computinginvolutionrepresentatives...");
invs:=Filtered(List(ConjugacyClasses(extG),Representative),x->Order(x)=2);;
Print(Size(invs),"involutionclasses\n");
ze-invs.gap");Read("analy

vs.gapquadrangle-in

decoGAPA.4.

Read("poly-utils.gap");#Somehelpercoder
group:automorphismfullCompute###1)DetermineapermutationpresentationofPGU(n,q)onourpoints
phi:=ActionHomomorphism(G,points,OnSubspacesByCanonicalBasis);
#2)Computeactionoffrobeniusautomorphism
gfrob:=Permutation(FrobeniusAutomorphism(F),points,OnTuplesTuples);
#3)Thefullautomorphismgroup:
Image(phi);:=GpermautG:=Group(Concatenation(GeneratorsOfGroup(Gperm),[gfrob]));
#Determinerepresentativesforallinvolutions
Print("Computinginvolutionrepresentatives...");
#ByAschbacher&Seitz,19.8,"InvolutionsinChevalleygroupsoverfieldsofevenorder",
#weknowforn=5andevenqthattherearethreeinvolutionclasses:
#Twoinnerones(determinedinapaperbyParkandYoo),plusafieldautomorphism.
ifn=5andIsEvenInt(q)then
inv1:=IdentityMat(n,F);inv1[1][4]:=One(F);inv1[2][5]:=One(F);
inv2:=IdentityMat(n,F);inv2[1][5]:=One(F);
invs:=[Image(phi,inv1),Image(phi,inv2),gfrob^(Order(gfrob)/2)];
#TODO:Dosomethingsimilarforn=4;andforoddq
else#Bydefault,weusebruteforcetofindallinvolutions
invs:=Filtered(List(ConjugacyClasses(autG),Representative),x->Order(x)=2);;
fi;Print(Size(invs),"involutionclasses\n");
psi:=ActionHomomorphism(autG,chambers,OnChambers);
(psi);Image:=ersautGOnChamb;Image(psi,invs):=invsze-invs.gap");Read("analy

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A.resultsterCompu

vs.gapanalyze-in#c_allAnalyze:=[];thec_distinvolutions:=[];
do[1..Size(invs)]inifortmp:=Print("Involinvs[i];ution",i,":");
ifelsetmpinPrint("polariautGOnChambersty;");thentmp:=Print("typetmp*tau;fi;preserving;");
tmp);tative(psi,PreImagesRepresen:=tmpifelsetmpinPrint("semiliGpermthennear;P");rint("linear;fi;");
#Sortc_all[i]all:=elementsList([0..MAXaccording_DIST],tox->[]);theirtheta-dist
forctmpin:=Distance(chambe[1..Size(chambers)]dors[c],chambers[c^invs[i]]);
chambers[c]);i][1+tmp],Add(c_all[od;");ta-dist...thealPrint("maximc_dist[i]Display(c_di:=st[i]);First(Reversed([0..MAX_DIST]),j->notIsEmpty(c_all[i][j+1]));
");geometry...ofelementsPrint("tmp:=Print(Size(Set(tmp,c_all[i][c_dist[ic->c[1])),]+1];"points,");
Print(Size(SPrint(Size(tmp),et(tmp,"chambc->c[2])),ers;");"lines,");
Print(NrComponentsChamberSet(tmp),"component(s)\n");
");nts...elemefixedPrint("tmp:=Print(Size(Sc_all[i][1];et(tmp,c->c[1])),"points,");
Print(Size(tPrint(Size(Smp),et(tmp,"chambc->c[2])),ers\n");"lines,");

od;

oly-utils.gappfunction(flag,g):=nonicalBasisOnFlagByCareturnList(flag,x->OnSubspacesByCanonicalBasis(x,g));
end;;c,g)function(:=OnChambersreturn[c[1]^g,Position(pointrow,OnSets(pointrow[c[2]],g))];
end;;componentconnectedCompute#s.ComponentsClocalcomps,hamberSetc,tmp;:=function(chambers)
#Listofcomponents.Eachcomponentisapairoflists.Thefirstcontains
#allpoints,thesecondalllinesinthatcomponent.
[];:=compsforcinchambersdo
tmp:=Filtered([1..Size(comps)],i->c[1]incomps[i][1]
comps[i][2]);inc[2]orifSize(tmp)=0then#Startanewcomponent
Add(comps,[[c[1]],[c[2]]]);
elifSize(tmp)=1then#Addtoexistingcomponent
);c[1]mps[tmp[1]][1],AddSet(coelse#AddSet(coMergemultiplemps[tmp[1]][2],componentsc[2]);
Assert(0,Size(tmp)=2);#Canonlybetwocomponents!
);[2]][1]comps[tmpcomps[tmp[1]][1],UniteSet(

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fi;

#RemoveUniteSet(comthesecondps[tmp[1]][2],componentcomps[tmp[2]][2]);
2]);tmp[,Remove(comps

od;comps;returnend;;

NrComponenreturntsChamberSetSize(Compon:=entsChamberSet(chambfunction(chambers)ers));
end;;

functiondistanceGeneric#c2),function(c1:=Distancethenc2=c1if0;returnelifc1[1]=c2[1]orc1[2]=c2[2]then
1;returnelifc1[1]inpointrow[c2[2]]orc2[1]inpointrow[c1[2]]then
2;returnelifc1[2]inpencil[c2[1]]orc2[2]inpencil[c1[1]]then
2;returnelifForAny(pointrow[c1[2]],p->pinpointrow[c2[2]])then
3;returnelifForAny(pencil[c1[1]],l->linpencil[c2[1]])then
3;returnelse4;returnfi;end;;

IsOpposite:=function(c1,c2)
returnDistance(c1,c2)=MAX_DIST;
end;;

A.4.

decoGAP

123

A.

124

terCompu

results

APPENDIXB

OBLEMSPROPEN

Inthefollowing,wepresentalistofopenproblemsandquestionsthataroseduring
thepreparationofthisthesis,roughlyorderedbythecorrespondingchapterand
section.

Flips(1)theAreassothereciatedexamplesbuilding,ofgroupswhichGcannotwithtbwineBliftedN-pairtoaBandNa-quasi-flipbuildingofGquasi-flip?(Seeof
Theorem2.2.2andthediscussionafterwards.)
(2)Involutionsofanon-sphericaltwinbuildingwhichdonotinterchangeitshalves
arenotquasi-flips.Theproblemisthatforthesemaps,thereseemstobeno
waytomakeuseoftheextrainformationprovidedbythetwinning.Hence
oneisreducedtothe(ratherbroad)theoryofgeneralbuildings.Buteven
foraninvolutoryautomorphismsofanarbitrarybuilding,onecanintroduce
thenotionofaθ-distance.Atleastforaffinebuildings,itseemspossibleto
deriveresultse.g.onθ-stabletwinapartments,byusingthesphericalbuilding
atinfinity,andsubsequentlydoublecosetdecompositions.
(3)ExtendtheparameterizationofthedoublecosetdecompositionGθ\G/Bfrom
Section2.7toGθ\G/PwherePisanarbitrary(spherical)parabolicsubgroup.
Thisshouldbestraightforward.Geometrically,onecouldarguewith(spherical)
residuesinsteadofchambers.BygoingfromaBorelsubgrouptoalarger
parabolicsubgroup,Gθ-orbitsmayfuse.Onewouldexpectthat
Gθ\G/P∼=WGθ(T)\WG(T)/WP(T)
∈TTwhereTisasetofrepresentativesoftheGθ×P-conjugacyclassesofθ-stable
tori.

125

problemsenOpB.

(4)Inthecaseofalgebraicgroupsoveralgebraicallyclosedfields,thereisaunique
“largest”(openanddense)orbitinGθ\G/B,the“bigcell”,seee.g.[HW93,
Sections4and9].ThishasbeenextendedtoKac-Moodygroupsincharacter-
[KW92].in0isticItwouldbeinterestingtostudythisforarbitraryKac-Moodygroups.The
weakZariskitopologyusedin[KW92]canbeextendedtoarbitraryKac-Moody
groups,buttheremanyofitsusefulpropertiesarenotknown.So,canoneuse
thisorsomeother“nice”topologysuchthatthereisauniqueopenanddense
orbit?Canoneunderstandorbitclosureinthistopology,andsaymeaningful
thingsaboutthegeneralorbitstructure?

Flipsinrank1and2
(1)StudyflipsofawiderclassofMoufangsets:E.g.allfiniteMoufangsets,or
evenallMoufangsetsoccurringin2-sphericalbuildings.Inparticular,their
transitivitypropertiesareofinterest:Bothtransitivityonthemovedchambers
(theflip-flopsystemoftheMoufangset)aswellasthefixedchambers.

systemsflip-flopofStructure(1)GivenaK-homogeneousquasi-flipθofanirreducibletwin-buildingoftype
(W,S),whatcanwesayaboutKinrelationtoS,otherthanthatitisspher-
ical?ForlinearflipsofalgebraicgroupsthisisansweredbyclassifyingSatake
diagrams.Forexample,aretherequasi-flipssuchthats∈Sexistswhere
K∪{s}isnotspherical?DoesthetheoryofSatakediagramsextendto(split)
groups?dyoKac-Mo(2)InProposition4.5.4weproveforK-homogeneousquasi-flipssatisfyingarank
2conditionthatCθisresiduallyconnectedif|K|≤2.CanthisboundonK
beimprovedorevendropped,possiblyafteraddingmorehypotheses?Also
(counter)exampleswouldbeofinterest,i.e.,quasi-flipsforwhichtherank2
conditionismet,yetCθisnotresiduallyconnected.
(3)AnanswertothefollowingquestionaboutCoxetersystemsθwouldgiveanaffir-
mativeanswertotheprecedingquestion,andimplythatCisalwaysresidually
connected:SupposewearegivenaCoxetersystem(W,S),anautomorphism
θof(W,S)oforderatmost2andasphericalandθ-invariantpropersubsetK
ofS.Moreoverforalls∈KwehaveswK=wKθ(s).
Thereisaposetstructureonthesetofallθ-twistedinvolutions(cf.[Spr84],
[RS90]):Startingwithaθ-twistedinvolutionw,givenagenerators∈S,then
exactlyoneofswandswθ(s)isaθ-twistedinvolutiondifferentfromw.We
writeswforthisandsetw<swifl(sw)<l(w),otherwisesw<w.

126

LetXbeasubsetofS,letK1,K2,K3besubsetsofK\XsuchthatK1∩K2=
K1∩K3=K2∩K3=∅.Supposewisaθ-twistedinvolutionabovewKinthe
posetwejustdescribed.IftherearedirectlyascendingchainsfromwKtow
insideeachofX∪Ki,canweprovethatthereissuchachaininsideX?
Ofcourse,ifanyoftheKiisempty(e.g.if|K|≤2),thenthisisobviously
true.SooneneedstodealwiththecasewhereallKiarenonempty.
(4)StudyconnectednessofRθanddirectdescentpropertiesofthoseMoufang
polygonswedidnotcoverinSection4.6,inparticularMoufanghexagons.
ThisissubjectofongoingresearchbyHendrikVanMaldeghemandtheauthor
[HVM].

Transitiveactionsonflip-flopsystems
(1)Arethereθquasi-flipsforwhichtheflip-flopsystemCθistheunionoftwoorθmore
distinctsetsCwasdefinedinSection5.1?(SeealsoRemark5.1.4.)SayC=
w∈XCwθ,thenwemusthavel(w1)=l(w2)foranyw1,w2∈X.Furthermore,as
aconsequenceofLemma2.3.4foreachw∈Xthereexistsaθ-stablespherical
subsetKwofSsuchthatwisthelongestelementinKw.Theresultsin
Chapter4furtherrestrictthepossiblediagramsoftheinvolvedbuilding,ifany
suchexampleevenexists.
(2)Arethereflip-floptransitiveflipswhicharenotdistancetransitive?
(3)Studytransitivitypropertiesofquasi-flipsofrank2buildings.Thismight
beeasier(andinsomecasesyieldmoreinsights)thantherank1(Moufang
set)case,andwouldstillallowtogivelocal-to-globaltransitivityresultsfor
quasi-flipsoftwo-sphericalbuildings.
(4)Findexamplesofflipsthatdonotallowuniformdescent(cf.Definition5.5.3).
Morespecifically,findexamplesthatadmitdirectdescentintoCθ(soifc∈/Cθ,
thenD(c)=∅),butnotuniformly.Givecriteriaastowhenaquasi-flipallows
(doesnotallow)uniformdescent.
Knowingmoreaboutthiswouldhelpansweringtheprecedingquestionon
transitivity(togetherwithknowledgeonthetransitivityinrank1and2).

problemsenopMore(1)LetGbealocallyfiniteKac-Moodygroupofnon-sphericaltype,θaquasi-flip,
andBaBorelgroup.ItisnotablethatBandGθhaveseveralinteresting
propertiesincommon:BothareingenerallatticesinacompletionofG(cf.
Theorem6.2.7,see[GM08]).ToBwecanassociatethechambersystemcop

127

problemsenOpB.

(2)

128

Cofθ;cforhambbotherswoppeareositeintheterestedchambinercconnectednessstabilizedbyandB,toGtransitivitθtheypropflip-floperties.system
ForBruhatbothwdecompegetosition)closelyandGrelated\G/Bdouble(cf.cosetSectiondecomp2.7)–inositionseachB\caseG/Bthe(theor-
bitscanbeparameterizedbyθ(quotientsof)Weylgroups.Thissimilarityis
notacoincidence:Fortheformer,thefactthatthetwinbuildingcanbecov-
foreredthebytlatterhecthathambtheersconbuildingtainedcaninbtewincoveredapartmebynthetscconhambtainingersofcallisθcen-stabletral,
twinapartmentsisrelevant(plusthefactthattwointersectingθ-stabletwin
apartmentsareGθ-conjugate).
Thefollowingquestioncomestomind:Canwegeneralizethistothestudy
ofdoublecosetspacesΓ\G/BwhereΓisanarbitrarylatticeinG?One
conideawtainingouldcbeontothetryonetohanddefineandaθsuitable-stabletwinreplacemenapartmentfortstheontthewinotherapartmenhand.ts

vForolutorymanyautomorphismsapplications,θoneandisinσ.Foterestedrinalgebraictheingroupsterplaythisofthaswobeencommresearcutinghedin-
toe.g.inKac-Mo[Hel88],odygroups[HW93],orev[HS01en][HS04].arbitraryItwgroupsouldbwithewaorthrootwhilegrouptodatum.extendthis

APPENDIXC

PHANTHEORYUSINGMOUFANGSETS

Inmanfoundationywaforyswhat[BS04],wecannobwecallconsideredPhantheotheryo.riginThere,oftheasptheoryecialofcaseflips,ofalaflipyingofthea
(spherical)buildingisdescribed(albeitindisguise,andtheterm“flip”isnoteven
used).ThegeometricsetupwasalreadybrieflysketchedinExample4.1.4.
Inloc.cit.,onlyfinitefieldsarecovered,usingcountingargumentswhichfailover
infinitefields.Inthisappendix,webrieflysketchhowthesecountingargumentscan
bereplacedbyMoufangsetargumentsasinSection3.3.1,e.g.Lemma3.3.5.We
provethefollowing:
TheoremC.1.LetCbeaMoufangtwinbuildingoftypeAn,n≥3,andassume
descthatentallpintoanelsCθcispontainossible.moreIfCthanθis10residualelements.lycLonneetθctebed,athenpropCerθisflipsimplyforcwhichonnedircteed.ct
θispNoteossiblethatifballyrotheotresultsgroupsinareChapteruniquely4,C2is-divisible,residuallyoriftheconnectedflipisandsemi-linear.directdescenThet
latteristhecasefortheflipusedin[BS04].
BytheclassificationofsphericalMoufangbuildings,itisknownthatanAnbuild-
ingforn≥3comesfroma(leftorright)vectorspaceoveraskewfieldK.Based
onthisknowledge,Tits’LemmaandTheoremC.1yieldapresentationofthegroup
SUn+1(K)asanamalgamofunitarysubgroupsSU2(K)andSU3(K).Aclassifica-
tiontheoremofofPhantypeAamalgamsnforSUovner+1(K).arbitraryAllin(skall,ew)wefieldsget:wouldthenimplyaPhan-type
CorollaryC.2.LetCbeaMoufangtwinbuildingoftypeAn,n≥3,definedovera
skewθfieldKwith|K|>10.Letθbeaproperflip.IfcharK=2orifθissemi-linear,
thenCisresiduallyconnectedandsimplyconnected.
TheremainderofthisappendixisdedicatedtoprovingTheoremC.1.Throughout,
CcorrespdenotesondingaCoMoufangxetersbuildingystemofofttypypeeAAnn.endoWewwilledwithadoptatheflipθ.viewLetof(aW,S)buildingbetheas

129

C.PhantheoryusingMoufangsets

anincidence(pre)geometryforourarguments(see[BC]or[Pas94]).Inparticular,
elementsoftype1and2ofthebuildingCwillbecalledpointsandlines,respectively
(i.e.,theycorrespondtochamberresiduesoftypeS\{1}andS\{2},respectively).
WedenotetheincidencepregeometrycomingfromCθbyGθ.1Itspointsandlines
formasubsetofthoseofthebuildingandtoavoidconfusionwewillrefertothemas
Gθ-pointsandGθ-lines.IftwoGθ-pointsarejoinedbyaGθ-line,wewillcallthemGθ-
collinear.NotethatwedropthetwinbuildingpointofviewandinsteadconsiderC
asaplainsphericalbuilding.Consequentlysinceθisaproperflip,itmapselements
oftypeitoelementsoftypen−i+1.

Flip-flopsystemscomingfromAnbuildings
θθwhicRecallhtwthatoptheointsarecollinearitadjacenytgraphwhenevΓassoertheyciatedarewithincidenCisttotheagraphcommononGGθ-p-line.ointsin
q2−Theqpoinremarktsofbcourseeforemak[BS04,esnoLemmasensefor2.2]infinitethatevfields.erylineWeofinsteadtheusegeometrythefolloconwing:tains
θGθLemma-lineformC.3.aprTheopperointsgener(ralizeesp.dlines)Moufangnotinsubset.Gofthepointrow(resp.pencil)ofa
Proof.Weargueforthepointrowofaline,thedualcaseissimilar.Applyθto
thepointsinthepointrowofaGθ-line,thenprojectthemback.Thisinducesa
permutationofthepointrowcompatiblewiththeMoufangstructureonit.The
pointsgeneralizedwhichareMoufangfixedsubsetareX.preciselySincethoseGθisnotainGgeometryθ.By(dueLemmatobeing3.3.4thisresiduallyisa
connected),thelineLcontainsaGθ-point,whenceXisapropersubset.
θθtheLemmaGθ-pointsC.4onL(Lemmawhich2.2areinGθlo-cc.olcit.)line.arIftoLpislieainGthe-linecandomplementpaGof-ptwooint,propthener
generθalizedMoufangθsubsets.Inparticular,ifLcontainsmorethan5pointsthenp
isG-collineartoaG-pointonL.
θθProaxioms,of.TtheaketwanoyGflags-line(p,LL)conand(tainingp,L)p,lietakineaanyGcommon-pointapaponrtmenL.tBy(annthe-simplex).building
Inparticular,piscollineartoeverypointonL,andeverylineinthepencilofθp
intersectsL.ApplyingLemmaC.3andprojectingsuitably,itfollowsthattheG-
pointsofLwhichareGθ-collineartoplieinthecomplementoftwopropergeneralized
Moufangsubsets.FinallyLemma3.3.7impliesthatiftherearemorethan5points
onL,thiscomplementisnon-empty.
ifFtheromnowunderlyingwewill(skew)assumefieldthatKallsatisfiespanels|K|con≥tain10.atleast11elements,whichfollows
1In[BS04],GθiscalledN

130

yconnectivitSimpleThefollowingisheavilybasedonSection3of[BS04],whereadescriptionofhomotopy
inincidencegeometryandsomeimportantLemmasaregiven;itisbesttoreadup
therefirstbeforeattemptingtounderstandthefollowing.Notethattheproofsof
Lemma3.2,Corollary3.3andLemma3.4inloc.cit.applyalmostverbatimlytoour
setup.2Therefore,wemerelyhavetoadaptLemmas3.5,3.6and3.7tocompletethe
C.1.TheoremofofproBeforewestart,weneedthefollowingauxiliaryresult.Weomittheproofwhich
issimilartothatofLemma3.3.7.
LemmaC.5.LetM(X,U)beaMoufangset,andletY1,Y2,Y3bethreepropergen-
eralizedMoufangsubsets.ThenX=Y1∪Y2∪Y3implies|X|≤10.
2NotethatfortheprojectivelineoverthefieldFq2,thisisequivalenttoasking
q+1>10,i.e.,q>3(thesameboundasin[BS04]).
Inthefollowing,weidentifylines,planes,hyperplanesetc.withthecollectionof
allpointsincidentwiththem.Thisiseasilyjustifiede.g.byidentificationofthe
elementsoftypeiwithpropersubspacesofdimensioniofKn+1.
Wecalltwopointsxandyperpendicularandwritex⊥yifx=projx,yθ(y),
equivalently,ifx∈θ(y).Sinceθisaninvolution,thisisclearlyasymmetricrelation.
Notethatx∈Gθifandonlyifx∈/θ(x).Thefollowingobservationisanimmediate
definition:theofconsequenceLemmaC.6.IftwoGθ-pointsareperpendicular,thentheyareGθ-collinear.
LemmaC.7(Lemma3.5inloc.cit.).EverytriangleinΓisdecomposable.
Proof.Letγ=abcabeatriangle(3-cycle)inΓ.IftheplaneU=a,b,cisinGθ
(i.e.,θ(U)∩U=0)thenγisgeometric.SosupposeU∈/Gθ.ThenU∩θ(U)must
beapointoutsideGθ(itcannotbealine,sinceUcontainsGθ-lines).Ifn≥4,we
canusethedirectdescentpropertytofindanelementinGθoftype≥4incidentto
U,whichthennecessarilyisincidenttoallofa,b,c,andhenceγisgeometric.
Thisleavesuswithn=3.Wewillfirstdealwiththecasewheretwopoints
onγ(say,aandb)areperpendicular.Inthiscasewesaythatγisofperptype.
LetW=θ(a,c),whichisalineofGθ.AnyGθ-pointd∈Wisbyconstruction
perpendiculartobothaandc,henceGθ-collineartoboth.ByLemmaC.4,the
pointsinthecomplementoftwopropergeneralizedMoufangsubsetsofWarealso
Gθ-collinearwithb.IfdisaGθ-pointonWthatisGθ-collinearwithbthenwesay
thatdisgoodifthetriangledbcdisgeometric,andthatitisbadotherwise.
2InofahLemmayperplane3.4,aissomewhatisomorphichtoiddenaflip-flopinductiongeometryargumenoftloiswerusedrank;whichohwevassumeser,thatthattheassumptionresidueis
notnecessary,residuallyconnectednessandasmallrefinementoftheproofsuffice.

131

C.PhantheoryusingMoufangsets

WIndeed,eifclaimdisthatbadthethenbadthepoinplanetsπformaspannedpropberyb,gec,disneralizednotmappMoufangedtoansubsetoppofositeW.
θpoinplanetbyoutsideθ,thatGθ.issNow:=sθ(∈π)θ(∩πb,cis)whicnon-empthisya.GθSince-line.πconButtainsthepGoints-lines,soutsidemustGθbeona
aGθ-lineformapropergeneralizedMoufangsubsetY.Sincethebaddbijectively
correspondtothesinY,theclaimfollows.
UsingLemmaC.5andthehypothesisthatallpanelscontainmorethan10ele-
ments,weconcludethatagoodpointdexists.Sinceaisperpendiculartobby
anassumptionelementofandGθd.byHenceabdaconstruction,isaandgeometricsinceb,dtriangle.isaGθSimilarly-line,theadcaplaneisa,b,geometric,dis
sincedisperpendiculartoaandc.Also,dbcdisgeometric,sincedisgood.Hence
γ=abcaisdecomposable.
Finally,letγ=abcabearbitrary.LetW=θ(a,c).ByLemmaC.4,thepoinθts
inthecomplementoftwopropergeneralizedMoufangsubsetsofWarealsoG-
andcollinearadcaarewithofb.perpLettdypbe,eonehenceofthesedecomppoinosablets.byThentheallabovthreee.Wetrianglesconcludeabdathat,dbcdall
osable.decompareγtrianglesLemmaC.8(Lemma3.6inloc.cit.).Every4-cycleinΓisdecomposable.
θPronoLof.whicLethγare=Gθabcdab-collinearea4to-cycle.bothBycandLemmadsimC.4ultaneously(resp.itsproformof),thetheGcomplemen-pointst
ofallthreepanelsproptoercongetainneralizedmorethanMoufang10pointssubsets.,weByconcludeLemmatheC.5existenceandsinceofawGeθ-poinassumedtp
θwhicNohwisitGfollows-collinearbyLetobmmaothcC.7andthatd.γisdecomposable,sinceitistheproductof
theshortercyclesapda,bcpbandcdpc.
LemmaC.9(Lemma3.7inloc.cit.).Every5-cycleinΓisdecomposable.
Proof.Letγ=abcdeabea5-cycle.ByLemmaC.4,disGθ-collineartosomeGθ-
pointponthelinea,b.
proNoductwitoffollothewsshorterbyLemmascyclesbcdpbC.7andandC.8apdea.thatγisdecomposable,sinceitisthe

132

[AB08][Abr96]G06][A[AM97][AS76][Asc77]VM99][A[Bae46][Bas93][BB05]

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yBibliograph

142

J-equivalent,5
µK-map,14-homogeneous,64
σ-twistedChevalleyinvolution,98
72-acute,θ21distance,-coθ72-orthogonal,θ73-parallel,θadjacenabsolutet,5element,77
7t,apartmen8win,tassociatedtwinbuilding,11
automorphismofofaaCoMoufangxeterset,system,142
11osition,decompBirkhoff9-pair,NB10win,t119,subgroup,BorelBruhatbuilding,6decomposition,9
7spherical,buildingtwin,7transitive,86
chambersystem,4
4ers,bhamcChevσa-tlleywisted,inv98olution,23
coconnected,distance,75,7

INDEX

Coconvxeterex,7,8group,1
1system,xeterCodescentDevillers-Mühlherrset,103property,30
directdistancedescenfunction,t,636
86e,transitivdistance31divisible,2condition,hangeExcflipBNbuilding,,2518
proplinear,er,10821,25
108semi-linear,30strong,45automorphism,flipflip-flopflip-flopsystem,geometry,47,6362
flip-flopinduced,transitiv62e,86
gallery,closed,5,85
5simple,groupgeneralizedofLietypMoufange,106subset,48
64homogeneous,HuaHuamap,subgroup,1414

143

Index

64connectedness,inherits3olution,vin3wisted,tisomorphism14sets,Moufangof15ted,oinp15e,isotop15,yisotopIwasawadecomposition,97,107,111
15isomorphism,Jordan110group,dyoKac-Mo112lattice,52,length,130lines,littleprojectivegroup,13
locallysplit,13
2t,elemenlongest13set,Moufang15ted,oinp47subset,Moufang2-spherical,n92map,normnumericalθ-codistance,21
61minimal,59flip,viousob87,osite,opp85,panel,parabolicsubgroup,2,9,11
9parallel,Phanchamber,21
21olution,vinPhan21residue,Phan21minimal,130ts,oinp19,yolaritp3t,otenprenilp87,jection,proprojectivecentralizer,44
6space,ejectivpro

144

erprop25-flip,NB21flip,buildingflipesequasi-flip,64,rank,2reduced,13reduction,5connected,residually85,residue,residuechambersystem,69
12GD-system,R12tered,cen12faithful,12reduced,2ot,ro3e,negativ3e,ositivp2simple,rootgroupdatum,12
13oups,grotrorootsubgroups,12
10saturated,72,spherical,43olution,vinstandard30quasi-flip,strong15group,structure6k,thic6thin,110functor,Tits9system,Titsetransitiv86building,86distance,twinflip-flop,apartmen86t,8
twinBN-pair,10
10saturated,7building,wint11ciated,assotwinTitssystem,10
88map,wistt

wistedt

wistedt

wistedt

e,ypt

2

uniform

action,

88

olution,vin

orbit,

88

t,descen

3

103

otenunip13radical,t

uniquely

eylW

-divisible,n

group,

9

31

Index

145

erdegangWherhaftlicWissensc

1980April24.

1993–1990

19931999–

1999

2005–2000

2005Okt.19.

Nov.2005–März2009

Nov.2005–Okt.2009

April2009

GeburtDarmstadtin

rmstadtaDinGymnasium

LahnsteininGymnasium

Abitur

StudiumderMathematikanderTUDarm-
stadt

DarmstadtTUMathematikdiplom

anPromotionsstudiumDarmstadtTUder

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ForschungamFachbereichMathematikderTU
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