FIVE LECTURES ON LATTICES IN SEMISIMPLE LIE GROUPS
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  • cours - matière potentielle : sur les reseaux des groupes de lie semisimples


FIVE LECTURES ON LATTICES IN SEMISIMPLE LIE GROUPS by Yves Benoist Abstract. — This text is an introduction to lattices ? in semisimple Lie groups G, in five independent lectures in which one answers to the following questions: Why do Coxeter groups give lattices in SO(p, 1) for p ≤ 9? Why do arithmetic constructions give lattices in SL(d,R) and SO(p, q)? Why do the unitary representations of G have an influence on the algebraic structure of ?? Why do the ?-equivariant factors of the Furstenberg boundary of G also have an influence on the algebraic structure of ?? Why does one need to study also lattices in semisimple Lie groups over local fields? Resume (Cinq cours sur les reseaux des groupes de Lie semisimples) Ce texte est une introduction aux reseaux ? des groupes de Lie semisimples G, en cinq cours independants dans lesquels on repond aux questions suivantes: Pourquoi les groupes de Coxeter donnent-ils des reseaux de SO(p, 1) pour p ≤ 9? Pourquoi les con- structions arithmetiques donnent-elles des reseaux de SL(d,R) et SO(p, q)? Pourquoi les representations unitaires de G ont-ils une influence sur la structure algebrique de ?? Pourquoi les facteurs ?-equivariants de la frontiere de Furstenberg de G ont-ils aussi une influence sur la structure algebrique de ?? Pourquoi doit-on ausi etudier les reseaux des groupes de Lie semisimples sur les corps locaux? Contents Introduction.

  • group

  • contents introduction

  • group ?

  • group has

  • groupe de coxeter

  • poincare's theorem

  • reseaux des groupes de lie semisimples sur les corps locaux

  • coxeter group

  • semisimple lie


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Nombre de lectures 79

Exrait

FIVELECTURESONLATTICESINSEMISIMPLELIE
GROUPS
ybYvesBenoist

Abstract
.—
ThistextisanintroductiontolatticesΓinsemisimpleLiegroups
G
,in
fiveindependentlecturesinwhichoneanswerstothefollowingquestions:WhydoCoxeter
groupsgivelatticesin
SO
(
p,
1)for
p

9?Whydoarithmeticconstructionsgivelatticesin
SL(
d,
R
)andSO(
p,q
)?Whydotheunitaryrepresentationsof
G
haveaninfluenceonthe
algebraicstructureofΓ?WhydotheΓ-equivariantfactorsoftheFurstenbergboundaryof
G
alsohaveaninfluenceonthealgebraicstructureofΓ?Whydoesoneneedtostudyalso
latticesinsemisimpleLiegroupsoverlocalfields?
Re´sume´
(Cinqcourssurlesre´seauxdesgroupesdeLiesemisimples)
Cetexteestuneintroductionauxre´seauxΓdesgroupesdeLiesemisimples
G
,en
cinqcoursinde´pendantsdanslesquelsonre´pondauxquestionssuivantes:Pourquoiles
groupesdeCoxeterdonnent-ilsdesre´seauxde
SO
(
p,
1)pour
p

9?Pourquoilescon-
structionsarithme´tiquesdonnent-ellesdesre´seauxdeSL(
d,
R
)etSO(
p,q
)?Pourquoi
lesrepre´sentationsunitairesde
G
ont-ilsuneinfluencesurlastructurealge´briquedeΓ?
PourquoilesfacteursΓ-e´quivariantsdelafrontie`redeFurstenbergde
G
ont-ilsaussiune
influencesurlastructurealge´briquedeΓ?Pourquoidoit-onausie´tudierlesre´seauxdes
groupesdeLiesemisimplessurlescorpslocaux?

Contents
Introduction.............................................................2
1.LectureonCoxeterGroups...........................................4
2.LectureonArithmeticgroups.........................................12
3.LectureonRepresentations...........................................24
4.LectureonBoundaries................................................35
5.LectureonLocalFields...............................................47
References..............................................................55

2000
MathematicsSubjectClassification
.—
11F06,20H10,22E40,22E46.
Keywordsandphrases
.—
lattices,Coxetergroups,arithmeticgroups,unitaryrepresentations,mix-
ing,propertyT,amenability,boundary,localfields.

2

YVESBENOIST

Introduction
ThistextisanintroductiontolatticesinsemisimpleLiegroups,infive
independentlectures.Itwasgivenduringthefirstweekofthe2004Summer
SchoolattheFourierInstituteinGrenoble.Wehopethatitwillattract
youngstudentstothistopicandconvincethemtoreadsomeofthemany
textbookscitedinthereferences.Weillustratefiveimportantmethodsof
thissubject:geometry,arithmetics,representations,boundaries,andlocal
fields.Oneforeachlecture.
AlatticeΓinarealsemisimpleLiegroup
G
isadiscretesubgroupforwhichthequotient
G/
Γsupportsa
G
-invariantmeasureoffinitevolume.OnesaysthatΓiscocompactif
thisquotientiscompact.WewilloftensupposethattheLiealgebra
g
issemisimple.This
isthecasefor
g
=
sl
(
d,
R
)or
g
=
so
(
p,q
).Thetwomainsourcesoflatticesare
-the
geometricmethod
:Oneconstructsaperiodictilingofthesymmetricspace
X
=
G/K
,where
K
isamaximalcompactsubgroupof
G
,withatile
P
offinitevolume.The
groupofisometriesofthistilingisthentherequiredlattice.Thisveryintuitivemethod,
initiatedbyPoincare´,seemstoworkonlyinlowdimension:evenifoneknowsbytheorical
argumentsthatitdoesexist,theexplicitdescriptionofsuchatile
P
inanydimensionis
stilladifficultquestion.Theaimofthefirstlectureistoconstructonefor
G
=SO(
p,
1),
where
p

9.
-the
arithmeticmethod
:Onethinksof
G
(orbetterofsomeproductof
G
byacompact
group)asbeingagroupofrealmatricesdefinedbypolynomialequationswithintegral
coefficients.ThesubgroupΓofmatriceswithintegralentriesisthenalatticein
G
.This
fact,duetoBorelandHarish-Chandra,impliesthat
G
alwayscontainsacocompactand
anoncocompactlattice.Theaimofthesecondlectureistoconstructsomeofthemfor
thegroups
G
=SL(
d,
R
)and
G
=SO(
p,q
).
AccordingtotheoremsofMargulisandGromov-Schoen,if
g
issimpleanddifferentfrom
so
(
p,
1)or
su
(
p,
1),thenalllatticesin
G
canbeconstructedbythearithmeticmethod.
When
g
=
so
(
p,
1)or
su
(
p,
1),quiteafewothermethodshavebeendevelopedinorderto
constructnewlattices.Eventhoughwewillnotdiscussthemhere,letusquote:

for
G
=SO(
p,
1):
-
p
=2:gluingtrousers(Fenchel-Nielsen);uniformization(Poincare´);
-
p
=3:gluingidealtetrahedraandDehnsurgery(Thurston);
-all
p
:hybridationofarithmeticgroups(Gromov,Piatetski-Shapiro).

for
G
=SU(
p,
1):
-
p
=2:groupsgeneratedbypseudoreflections(Mostow);fundamentalgroupofalgebraic
surfaces(Yau,Mumford);

3:omudilFIVELECTURESONLATTICESINSEMISIMPLELIEGROUPS
3

psaecsofewgithdeponitsontehilen;oholonymgruospofolaclystsmesp-≤(Deligne,Mostow,Thurston).
-all
p
:unknownyet.
Oneofthemainsuccessesofthetheoryoflatticesisthatitgaveinaunifiedwaymany
newpropertiesofarithmeticgroups.OnedoesnotusethewayinwhichΓhasbeen
constructedbutjusttheexistenceofthefiniteinvariantmeasure.Akeytoolisthetheory
ofunitaryrepresentations,andmorepreciselytheasymptoticbehaviorofcoefficientsof
vectorsinunitaryrepresentations.Wewillexplainthisinthethirdlecture.
AnotherimportanttoolaretheboundariesassociatedtoΓ.Wewillseeinthefourth
lecturehowtheyareusedintheproofoftheMargulisnormalsubgrouptheorem,which
saysthat
latticesinrealsimpleLiegroupsofrealrankatleast
2
arequasisimple
,i.e.their
normalsubgroupsareeitherfiniteoroffiniteindex.
Thegeneraltheorywedescribe

dsofargivesinformationonarithmeticgroupslike
SL(
d,
Z
),SO(
d,
Z
[
i
]),orSp(
d,
Z
[2]).Itca

nbeextendedto
S
-arithmeticgroupslike
SL(
d,
Z
[
i/N
]),SO(
d,
Z
[1
/N
]),orSU(
p,q,
Z
[2
/N
])...Theonlythingonehastodois
toreplacetherealLiegroup
G
byaproductofrealand
p
-adicgroups.Theaimofthe
lastlectureistoexplainhowtoadapttheresultsofthepreviouslecturestothatsetting.
Forinstance,wewillconstructcocompactlatticesinSL(
d,
Q
p
)andseethattheyare
quasisimplefor
d

3.
Thistextisslightlylongerthantheorallecture,parcequ’autableauilestplusfacilede
remplacerunede´monstrationtechniqueparunmagnifiquecrobard,unprincipege´ne´ral,
unexempleinsignifiant,unexerciceintordablevoireunegrimacee´vocatrice.Onefor
eachlecture.Nevertheless,therearestillmanyimportantclassicalthemesinthissubject
whichwillnotbediscussedhere.Letusjustquoteafew:cohomologicaldimension
andcohomology,universalextensionandthecongruencesubgroupproperty,rigidityand
superigidity,countingpointsandequirepartition,Shimuravarieties,quasiisometries...
Ungrandmerciauxauditeursdel’E´coled’e´te´quiparleurscritiquesm’ontpermis
d’ame´liorercetexte:NirAvni,UriBader,PierreEmmanuelCaprace,YvesdeCornulier,
DamienFerte´,FrancoisGue´ritaud,FrancoisMaucourant,BarbaraSchapira,etaussiGae-
tanChenevier,FannyKassel,VincentLafforgue,BertrandRemyetlereferee.

Foranundergraduateintroductiontotilingsandlattices,onecanread[2].

4

YVESBENOIST

1.LectureonCoxeterGroups
Inthefirstlecture,weconstructafewlatticesin
SO
(
p,
1)bythegeometric
method,when
p

9.
1.1.Introduction.—
Thegeometricmethodofconstructionoflatticeshasbeenini-
tiatedbyPoincare´in1880.Inhisconstruction,thegroup
G
isthegroup
PO
+
(2
,
1)of
isometriesofthehyperbolicplane
H
2
.Onebeginswithapolygon
P

H
2
andwitha
familyofisometrieswhichidentifytheedgesof
P
twobytwo.Whentheseisometries
satisfysomecompabilityconditionssayingthat“thefirstimagesof
P
giveatilingaround
eachvertex”,thePoincare´theoremsaysthatthegroupΓgeneratedbytheseisometries
actsproperlyon
H
2
,with
P
asafundamentaldomain.Inparticular,when
P
hasfinite
volume,thegroupΓisalatticein
G
.
Thereexistsahigher-dimensionalextensionofPoincare´’stheorem.Onereplaces
H
2
dbythe
d
-dimensionalhyperbolicspace
H
,thepolygon
P
byapolyhedron,theedges
bythe(
d

1)-faces,andtheverticesbythe(
d

2)-faces(see[
16
]).Inmostofthe
explicitly-knownexamples,onechoosesΓtobegeneratedbythesymmetrieswithrespect
tothe(
d

1)-facesof
P
.Theaimofthislectureistopresentaproof,duetoVinberg,
ofthisextensionofPoincare´’stheoremandtodescribesomeoftheseexplicitpolyhedra
for

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