FIVE LECTURES ON LATTICES IN SEMISIMPLE LIE GROUPS

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