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FIVE LECTURES ON LATTICES IN SEMISIMPLE LIE GROUPS

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55 pages

  • 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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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
d

9.Inthiscase,thegroupΓisaCoxetergroup.Asaby-product,wewillobtain
geometricproofsofsomeofthebasicpropertiesofCoxetergroups.
Eventhoughthegeometricconstructionmayseemlessefficientthanthearithmetic
one,itisstillanimportanttool.
1.2.Projectivetransformations.—
Letusbeginwithafewbasicdefinitionsand
properties.Let
V
:=
R
d
+1
,
S
d
=
S
(
V
):=(
V

0)
/
R
+
×
betheprojectivesphere,and
SL
±
(
d
+1
,
R
)bethegroupofprojectivetransformationsof
S
d
.
Definition1.1
.—
Areflection
σ
isanelementoforder
2
of
SL
±
(
d
+1
,
R
)
whichisthe
identityonanhyperplane.Allreflectionsareoftheform
σ
=
σ
α,v
:=
Id

α

v
forsome
α

V

and
v

V
with
α
(
v
)=2
.
-Arotation
ρ
isanelementof
SL
±
(
d
+

1
,
R
)
whichis

theidentityonasubspaceof
codimension
2
andisgivenbyamatrix
cos
θ

sin
θ
inasuitablesupplementary
sin
θ
cos
θ
basis.Thereal
θ

[0

]
istheangleoftherotation.
Let
σ
1
=
σ
α
1
,v
1
,
σ
2
=
σ
α
2
,v
2
betwodistinctreflections,Δbethegrouptheygenerate,
a
12
:=
α
1
(
v
2
),
a
21
:=
α
2
(
v
1
),and
L
:=
{
x

S
d

1
|
x

0

2
|
x

0
}
.Thefollowing
elementarylemmatellsuswhentheimages
δ
(
L
),
δ

Δ,tileasubset
C
of
S
d
,i.e.when
◦theinteriors
δ
(
L
),
δ

Δ,aredisjoints.Theset
C
isthentheunion
C
=
δ

Δ
δ
(
L
).
SLemma1.2
.—
a)If
a
12
>
0
or
a
21
>
0
,the
δ
(
L
)
,
δ

Δ
,donottile(anysubsetof
S
d
.
b)Supposenow
a
12

0
and
a
21

0
.Considerthefollowingfourcases:

FIVELECTURESONLATTICESINSEMISIMPLELIEGROUPS
5

b1)
a
12
a
21
=0
.Ifboth
a
12
and
a
21
areequalto
0
,thentheproduct
σ
1
σ
2
isoforder
2
,the
group
Δ
is
Z
/
2
×
Z
/
2
,andthe
δ
(
L
)
,
δ

Δ
,tile
S
d
.Otherwisetheydonottile.
2b2)
0
<a
12
a
21
<
4
.Theproduct
σ
1
σ
2
isarotationofangle
θ
givenby
4cos(
θ/
2)=
a
12
a
21
.
If
θ
=2
π/m
forsomeinteger
m

3
then
σ
1
σ
2
isoforder
m
,thegroup
Δ
is
Z
/
2
×
Z
/m
,
andthe
δ
(
L
)
,
δ

Δ
,tile
S
d
.Otherwisetheydonottile.
b3)
a
12
a
21
=4
.Theproduct
σ
1
σ
2
isunipotentandthe
δ
(
L
)
,
δ

Δ
,tileasubset
C
of
S
d
whoseclosureisahalf-sphere.
b4)
a
12
a
21
>
4
.Theproduct
σ
1
σ
2
hastwodistinctpositiveeigenvaluesandthe
δ
(
L
)
,
δ

Δ
,tileasubset
C
of
S
d
whoseclosureistheintersectionoftwodistincthalf-spheres.
Proof
.—Thislemmareducestoa2-dimensionalexercisethatislefttothereader.

α
(
v
)
α
(
v
)
Remark
Thecross-ratio[
α
1

2
,v
1
,v
2
]:=
α
11
(
v
12
)
α
22
(
v
21
)
=
a
12
4
a
21
isaprojectiveinvariant.
1.3.Coxetersystems.—
A
Coxetersystem
(
S,M
)isthedataofafiniteset
S
and
amatrix
M
=(
m
s,t
)
s,t

S
withdiagonalcoefficients
m
s,s
=1andnondiagonalcoefficients
m
s,t
=
m
t,s
∈{
2
,
3
,...,
∞}
.Thecardinal
r
of
S
iscalledthe
rank
oftheCoxetersystem.
TosuchaCoxetersystemoneassociatesthecorresponding
Coxetergroup
W
=
W
S
definedbythesetofgenerators
S
andtherelations(
st
)
m
s,t
=1,forall
s,t

S
such
that
m
s,t
6
=

.For
w
in
W
,thelength

(
w
)isthesmallestinteger

suchthat
w
isthe
productof

elementsof
S
.
ACoxetergrouphasanatural
r
-dimensionalrepresentation
σ
S
,calledthe
geometric
representation
,whichisdefinedinthefollowingway.Let(
e
s
)
s

S
bethecanonicalbasis
of
R
S
.The
Titsform
on
R
S
isthesymmetricbilinearformdefinedby
πB
S
(
e
s
,e
t
):=

cos(
m
s,t
)forall
s,t

S.
AccordingtoLemma1.2,theformula
σ
S
(
s
)
v
=
v

2
B
S
(
e
s
,v
)
e
s

s

S,v

E
S
definesamorphism
σ
S
of
W
intotheorthogonalgroupoftheTitsform.Let
P
S
bethe
simplexinthesphere
S
r

1
ofthedualspacedefinedby
P
S
:=
{
f

S
r

1
/f
(
e
s
)

0
/

s

S
}
.
AsaspecialcaseoftheVinbergtheoremstatedinthenextsection,wewillseethe
followingtheorem,duetoTits.
Theorem1.3
.—
(Tits)
Therepresentation
σ
S
isfaithful,itsimage
Γ
S
isdiscreteand
thetranslates
t
γ
(
P
S
)
,
γ

Γ
S
,tileaconvexsubset
C
S
ofthesphere
S
r

1
.
Remarks
-Theconvex
C
S
iscalledthe
Titsconvexset
.
-ForafewCoxetergroupswith
r

10,called
hyperbolic
,wewillprovethattheTitsform
isLorentzianofsignature(
r

1
,
1)andthatthegroupΓ
S
isalatticeinthecorresponding
orthogonalgroup.

6

YVESBENOIST

Corollary1.4
.—
Foreverysubset
S


S
,thenaturalmorphism
ρ
S,S

:
W
S


W
S
is
injective.
ProofofCorollary1.4
.—Therepresentation
σ
S

isequaltotherestrictionof
σ
S

ρ
S,S

tothevectorspace
<e
s
,s

S

>
.
1.4.Groupsofprojectivereflections.—
Inthissectionwestudygroupsgeneratedbyprojectivereflectionsfixing
thefacesofsomeconvexpolyhedron
P
ofthesphere
S
d
.
Let
P

S
d
bea
d
-dimensional
convexpolyhedron
,i.e.theimagein
S
d
ofaconvex
polyhedralconeof
R
d
+1
with0omitted.A
k
-face
of
P
isa
k
-dimensionalconvexsubset
of
P
obt

ainedasanintersectionof
P
withsomehypersphereswhichdonotmeetthe
interior
P
.A
face
isa(
d

1)-faceandanedgeisa0-face.
Let
S
bethesetoffacesof
P
andforevery
s
in
S
,onechoosesaprojectivereflection
σ
s
=
Id

α
s

v
s
with
α
s
(
v
s
)=2whichfixes
s
.Asuitablechoiceofsignsallowsusto
supposethat
P
isdefinedbytheinequalities(
α
s

0)
s

S
.Let
a
s,t
:=
α
s
(
v
t
)for
s,t

S
.
LetΓbethegroupgeneratedbythereflections
σ
s
.
AccordingtoLemma1.2,ifwewanttheimages
γ
(
P
)totilesomesubsetof
S
d
,the
followingconditionsarenecessary:forallfaces
s,t

S
suchthattheintersection
s

t
is
a(
d

2)-dimensionalfaceof
P
,onehas

(1)
a
s,t

0and(
a
s,t
=0
⇐⇒
a
t,s
=0)
π(2)
a
s,t
a
t,s

4or
a
s,t
a
t,s
=4cos
2
(
m
s,t
)with
m
s,t
integer
,m
s,t

2
Conversely,thefollowingtheoremstatesthattheseconditionsarealsosufficient.
Let(
S,M
)betheCoxetersystemgivenbytheseintegers
m
s,t
andcompletedby
m
s,t
=

wheneither
s

t
=

,codim(
s

t
)
6
=2,or
a
s,t
a
t,s

4.Notethat,whenthepolyhedron
isthesimplex
P
S
oftheprevioussection,theCoxetersystemistheonewestartedwith.
Theorem1.5
.—
(Vinberg)
Let
P
beaconvexpolyhedronof
S
d
and,foreachface
s
of
P
,let
σ
s
=
Id

α
s

v
s
beaprojectivereflectionfixingtheface
s
.Supposethatconditions
(1)and(2)aresatisfiedforevery
s,t
suchthat
codim(
s

t
)=2
.Let
Γ
bethegroup
generatedbythereflections
σ
s
.Then
(a)thepolyhedra
γ
(
P
)
,for
γ
in
Γ
,tilesomeconvexsubset
C
of
S
d
;
(b)themorphism
σ
:
W
S

Γ
givenby
σ
(
s
)=
σ
s
isanisomorphism;
(c)thegroup
Γ
isdiscretein
SL
±
(
d
+1
,
R
)
.
Inotherwords,tobesurethataconvexpolyhedronanditsimagesbyagroupgenerated
byprojectivereflectionsthroughitsfacestilesomepartofthesphere,itisenoughtocheck
localconditions“aroundeach2-codimensionalface”.
Wewillstillcall
C
the
Titsconvexset
.Itmaynotbeopen.

FIVELECTURESONLATTICESINSEMISIMPLELIEGROUPS
7

Remark
TheproofofTheorem1.3givenbyTitsin[
8
]canbeadaptedtogetTheorem
1.5(see[
24
],Lemma1).Inthislecture,wewillfollowVinberg’sproof,whichismore
geometric.
1.5.Theuniversaltiling.—
ToproveTheorem1.5,oneintroducesanabstractspace
X
obtainedby
gluingcopiesof
P
indexedbytheCoxetergroup
W
:=
W
S
alongtheirfaces
andoneprovesthatthisspaceisconvex.
Formally,onedefines
X
:=
W
×
P/

wheretheequivalencerelation

isgeneratedby
(
w,p
)

(
w

,p

)
⇐⇒∃
s

S/w

=
ws
and
p

=
p
=
σ
s
(
p
)
.
Onedenotesby
P
sing
theunionofthe3-codimensionalfacesof
P
,andsets
P
reg
=
P

P
sing
,
X
sing
:=
W
×
P
sing
/

,
X
reg
=
X

X
sing
.TheCoxetergroup
W
actsnaturally
on
X
andon
S
d
.Let
π
:
X

S
d
bethemapdefinedby
π
(
w,p
):=
wp
.
Lemma1.6
.—
a)For
x
in
P
,let
W
x

W
bethesubgroupgeneratedby
σ
s
for
s

x
.
Then
V
x
:=
W
x
×
P/

isaneighborhoodof
x
in
X
.
b)Themap
π
is
W
-equivariant,i.e.

w

W
,

x

X,π
(
wx
)=

(
x
)
.
c)Forall
x
in
X
reg
,thereexistsaneighborhood
V
x
of
x
in
X
suchthat
π
|
V
x
isahomeo-
morphismontoaconvexsubsetof
S
d
.
Proof
.—a)Let
P
x
beanopenneighborhoodof
x
in
P
whichdoesnotmeetthefacesof
P
notcontaining
x
.Then,
W
x
×
P
x
/

isopenin
X
.
b)Easy.
c)Thisisaconsequenceofa),b),lemma1.2andofhypotheses(1)and(2).

Asegmenton
S
d
isa1-dimensionalconvexsubsetwhichisnotacircle.Letustransfer
thisnotionofsegmentto
X
.
Definition1.7
.—
Forevery
x
,
y
in
X
,asegment
[
x,y
]
isacompactsubsetof
X
suchthattherestrictionof
π
to
[
x,y
]
isahomeomorphismontosomesegmentof
S
d
with
end-points
π
(
x
)
and
π
(
y
)
.
Wedonotknowyetthatsuchasegmentdoesexist.Itispreciselywhatwewantto
shownow.

Letusdenoteby
∂P
=
P

P
theunionofthefacesof
P
and
∂X
:=
W
×
∂P/

.The
followinglemmaisthekeylemma.Foreachpoint
z
in
P
reg
onedefinesitsmultiplicity
by
m
(
z
):=
m
s,t
if
z

s

t
forsome
s
6
=
t
,andby
m
(
z
):=
1
∂P
(
z
)otherwise.Weextend
thisfunctionon
X
reg
bytheformula
m
(
wz
):=
m
(
z
).
◦◦Lemma1.8
.—
Fix
w

W
.Let
S
=
S
w
bethesetofall
(
x,y
)

P
×
P

X
×
X
such
that
π
(
x
)
6
=

π
(
y
)
,andsuchthatthesegment
[
x,wy
]
existsandiscontainedin
X
reg
.
Suppose
S
6
=

.Then

YVESBENOIST

8a)thesum
m
(
z
)
isaconstant
L
(
w
)
on
S
dependingonlyon
w
;
Xz

[
x,wy
]
b)theset
S
isdensein
P
×
P
.
Theabovesumcountsthenumberoffacescrossedbythesegment[
x,wy
].Wewillsee
laterthatthisnumber
L
(
w
)isequaltothelength

(
w
).

Proof
.—Let
L
(
x,y,w
)betheabovesum.Accordingtothelocalanalysisgivenin
Lemma1.2,whenthesegment[
x,wy
]crossestheinteriorofa2-codimensionalface
w

(
s

t
),onehas
m
s,t
<

.Moreover,thislocalanalysisprovesthatthefunction(
x,y
)

L
(
x,y,w
)islocallyconstant(thisisthemainpointinthisproof,seetheremarkbelow).
Choose
L

0suchthattheset
S
L
:=
{
(
x,y
)

S/L
(
x,y,w
)=
L
}
isnonempty.One
◦◦knowsthat
S
L
isopenin
P
×
P
.Noticethat,for(
x,y
)in
S
L
,theonlytiles
w

P

X
crossedbythesegment[
x,wy
]satisfy

(
w

)

L
,theybelongtoafixedfinitesetoftiles.
andisincludedinthecompact

(
w

)

L
w

(
P
).Moreover,since
P
sing
isofcodimension3,
So,byacompactnessargument,
S
forany(
x,y
)intheclosure
S
L
,thesegment[
x,wy
]exists
removingsomesubsetofcodimension2in
S
,onecanfindanopen,connected,anddense
◦◦subset
S

of
P
×
P
suchthat
S
L

S


S
L
.Hence,successively,
S
L

S

isopenandclosed
◦◦in
S

,
S

isincludedin
S
L
,
S
L
isdensein
P
×
P
,and
S
L
=
S
.
Thenextstatementisacorollaryofthepreviousproof.
Lemma1.9
.—
Forevery
x,x

in
X
,thereexistsatleastonesegment
[
x,x

]
joining
.mehtMoreover,when
π
(
x
)
6
=

π
(
x

)
,thissegmentisunique.
Proof
.—Keepnotationsfromthepreviouslemmawith
x

=
wy
.
Weknowtheimplication
S
w
6
=

=

S
w
=
P
×
P
.Thisallowstoprovebyinduction
on

(
w
)that
S
w
6
=

,bylettingthepoint
y
movecontinuouslythroughaface.The
uniquenessfollowsfromtheuniquenessofthesegmentjoiningtwonon-antipodalpoints
onthesphere
S
d
.
Lemma1.10
.—
Themap
π
:
X

C
isbijectiveand
C
isconvex.
Proof
.—Let
x,x

betwopointsof
X
.AccordingtoLemma1.9,thereisasegment[
x,x

]
joiningthem.Henceif
π
(
x
)=
π
(
x

),onemusthave
x
=
x

.Thisprovesthat
π
:
X

C
isbijective.Twopointsof
C
canalsobejoinedbyasegment,hence
C
isconvex.

ProofofTheorem1.5
.—(a),(b)followfromLemma1.10,and(c)followsfrom(a).

Remark
LetuspointouthowcrucialLemma1.8is.ConsiderthefollowinggroupΓ
generatedbytwolineartransformations
g
1
and
g
2
of
R
2
,whichidentifytheoppositefaces

FIVELECTURESONLATTICESINSEMISIMPLELIEGROUPS

9

ofaconvexquadrilateral
P
:
-
g
1
isthehomothetyofratio2,
-
g
2
isarotationwhoseangle
α/π
irrational,and
2x-
P
:=
{
(
x,y
)

R
2
/
1

x

2and

y