Table Of ContentIntroduction to Arithmetic Groups
Preliminaryversion0.5(April6,2008)
SendcommentstoDave.Morris@uleth.ca
Dave Witte Morris
DepartmentofMathematicsandComputerScience
UniversityofLethbridge
Lethbridge,Alberta,T1K3M4,Canada
Dave.Morris@uleth.ca,http://people.uleth.ca/~dave.morris/
April 7, 2008
Copyright(cid:13)c 2001–2008DaveWitteMorris.Allrightsreserved.
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i
Acknowledgments
Inwritingthisbook,IhavereceivedmajorhelpfromdiscussionswithScot
Adams, S.G. Dani, Benson Farb, G.A. Margulis, Gopal Prasad, M.S. Raghu-
nathan, T. N. Venkataramana, and Robert J. Zimmer. I have also benefited
from the comments and suggestions of many other colleagues, includ-
ingMarcBurger,IndiraChatterji,AlessandraIozzi,AndersKarlsson,Sean
Keel, Nicolas Monod, Hee Oh, Alan Reid, Yehuda Shalom, Shmuel Wein-
berger, Barak Weiss, and Kevin Whyte. They may note that some of the
remarkstheymadetomehavebeenreproducedherealmostverbatim.
I would like to thank É. Ghys, D. Gaboriau and the other members of
the École Normale Supérieure of Lyon for the stimulating conversations
and invitation to speak that were the impetus for this work, and B. Farb,
A.EskinandtheothermembersoftheUniversityofChicagomathematics
department for encouraging me to write this introduction, for the oppor-
tunitytolecturefromit,andforprovidingastimulatingaudience.
I am also grateful to the University of Chicago, the École Normale
Supérieure of Lyon, the University of Bielefeld, the Isaac Newton Institute
for Mathematical Sciences, the University of Michigan, and the Tata Insti-
tuteforFundamentalResearchfortheirwarmhospitalitywhilevariousof
the chapters were being written. The preparation of this manuscript was
partially supported by research grants from the National Science Founda-
tionoftheUSAandtheNationalScienceandEngineeringResearchCouncil
ofCanada.
iii
List of Chapters
Part I Geometric Motivation
1 WhatisaLocallySymmetricSpace? 3
2 GeometricMeaningofR-rankandQ-rank 19
Part II Fundamentals
3 IntroductiontoSemisimpleLieGroups 27
4 BasicPropertiesofLattices 41
5 WhatisanArithmeticLattice? 71
6 ExamplesofLattices 95
Part III Important Concepts
7 RealRank 125
8 Q-Rank 139
9 ErgodicTheory 157
10 AmenableGroups 167
11 Kazhdan’sProperty(T) 189
Part IV Major Results
12 MargulisSuperrigidityTheorem 199
13 NormalSubgroupsof 217
Γ
14 ReductionTheory:AFundamentalSetforG/
Γ
hnotwrittenyeti 231
15 ArithmeticLatticesinClassicalGroups 233
Appendices
A AssumedBackground 269
B WhichClassicalGroupsareIsogenous? 283
C CentralDivisionAlgebrasoverNumberFields 291
iv
Contents
Part I Geometric Motivation 1
Chapter1 WhatisaLocallySymmetricSpace? 3
§1A. Symmetricspaces . . . . . . . . . . . . . . . . . . . . . . . . . 3
§1B. Howtoconstructasymmetricspace . . . . . . . . . . . . . 7
§1C. Locallysymmetricspaces . . . . . . . . . . . . . . . . . . . . 11
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter2 GeometricMeaningofR-rankandQ-rank 19
§2A. Rankandrealrank . . . . . . . . . . . . . . . . . . . . . . . . 19
§2B. Q-rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Part II Fundamentals 25
Chapter3 IntroductiontoSemisimpleLieGroups 27
§3A. Thestandingassumptions . . . . . . . . . . . . . . . . . . . . 27
§3B. Isogenies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
§3C. WhatisasemisimpleLiegroup? . . . . . . . . . . . . . . . . 28
§3D. ThesimpleLiegroups . . . . . . . . . . . . . . . . . . . . . . 31
§3E. G isalmostZariskiclosed . . . . . . . . . . . . . . . . . . . . 35
§3F. Jacobson-MorosovLemma . . . . . . . . . . . . . . . . . . . . 39
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter4 BasicPropertiesofLattices 41
§4A. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
§4B. Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . 45
§4C. Irreduciblelattices . . . . . . . . . . . . . . . . . . . . . . . . . 46
§4D. Unboundedsubsetsof \G . . . . . . . . . . . . . . . . . . . 48
Γ
§4E. Intersectionof withothersubgroupsofG . . . . . . . . . 51
Γ
§4F. BorelDensityTheoremandsomeconsequences . . . . . . 52
§4G. ProofoftheBorelDensityTheorem . . . . . . . . . . . . . . 54
§4H. isfinitelypresented . . . . . . . . . . . . . . . . . . . . . . . 57
Γ
§4I. hasatorsion-freesubgroupoffiniteindex . . . . . . . . 60
Γ
v
vi CONTENTS
§4J. hasanonabelianfreesubgroup . . . . . . . . . . . . . . . 65
Γ
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Chapter5 WhatisanArithmeticLattice? 71
§5A. Definitionofarithmeticlattices. . . . . . . . . . . . . . . . . 71
§5B. MargulisArithmeticityTheorem . . . . . . . . . . . . . . . . 76
§5C. Unipotentelementsofnoncocompactlattices . . . . . . . . 79
§5D. Howtomakeanarithmeticlattice . . . . . . . . . . . . . . . 81
§5E. Restrictionofscalars . . . . . . . . . . . . . . . . . . . . . . . 83
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter6 ExamplesofLattices 95
§6A. ArithmeticlatticesinSL(2,R) . . . . . . . . . . . . . . . . . . 95
§6B. TeichmüllerspaceandmodulispaceoflatticesinSL(2,R) 100
§6C. ArithmeticlatticesinSO(1,n) . . . . . . . . . . . . . . . . . 100
§6D. SomenonarithmeticlatticesinSO(1,n) . . . . . . . . . . . 105
§6E. NoncocompactlatticesinSL(3,R) . . . . . . . . . . . . . . . 113
§6F. CocompactlatticesinSL(3,R) . . . . . . . . . . . . . . . . . 117
§6G. LatticesinSL(n,R) . . . . . . . . . . . . . . . . . . . . . . . . 120
§6H. QuaternionalgebrasoverafieldF . . . . . . . . . . . . . . . 121
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Part III Important Concepts 123
Chapter7 RealRank 125
§7A. R-splittori. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
§7B. Definitionofrealrank . . . . . . . . . . . . . . . . . . . . . . 126
§7C. Relationtogeometry . . . . . . . . . . . . . . . . . . . . . . . 128
§7D. Parabolicsubgroups. . . . . . . . . . . . . . . . . . . . . . . . 129
§7E. Groupsofrealrankzero . . . . . . . . . . . . . . . . . . . . . 132
§7F. Groupsofrealrankone . . . . . . . . . . . . . . . . . . . . . 134
§7G. Groupsofhigherrealrank. . . . . . . . . . . . . . . . . . . . 136
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Chapter8 Q-Rank 139
§8A. Q-splittori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
§8B. Q-rankofanarithmeticlattice . . . . . . . . . . . . . . . . . 141
§8C. IsogeniesoverQ . . . . . . . . . . . . . . . . . . . . . . . . . . 142
§8D. Q-rankofanylattice . . . . . . . . . . . . . . . . . . . . . . . 144
§8E. ThepossibleQ-ranks . . . . . . . . . . . . . . . . . . . . . . . 144
§8F. LatticesofQ-rankzero . . . . . . . . . . . . . . . . . . . . . . 146
vii
§8G. LatticesofQ-rankone . . . . . . . . . . . . . . . . . . . . . . 148
§8H. LatticesofhigherQ-rank . . . . . . . . . . . . . . . . . . . . . 148
§8I. ParabolicQ-subgroups . . . . . . . . . . . . . . . . . . . . . . 150
§8J. Thelarge-scalegeometryof \X . . . . . . . . . . . . . . . . 152
Γ
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Chapter9 ErgodicTheory 157
§9A. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
§9B. Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
§9C. Consequencesofaninvariantprobabilitymeasure. . . . . 161
§9D. ProofoftheMooreErgodicityTheorem (optional) . . . . . . 163
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Chapter10 AmenableGroups 167
§10A. Definitionofamenability . . . . . . . . . . . . . . . . . . . . . 167
§10B. Examplesofamenablegroups . . . . . . . . . . . . . . . . . 168
§10C. Othercharacterizationsofamenability . . . . . . . . . . . . 171
§10D. Somenonamenablegroups . . . . . . . . . . . . . . . . . . . 182
§10E. Closedsubgroupsofamenablegroups . . . . . . . . . . . . 184
§10F. EquivariantmapsfromG/P toProb(X). . . . . . . . . . . . 186
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Chapter11 Kazhdan’sProperty(T) 189
§11A. Kazhdan’sproperty(T) . . . . . . . . . . . . . . . . . . . . . 189
§11B. SemisimplegroupswithKazhdan’sproperty . . . . . . . . 191
§11C. LatticesingroupswithKazhdan’sproperty . . . . . . . . . 191
§11D. FixedpointsinHilbertspaces . . . . . . . . . . . . . . . . . . 193
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Part IV Major Results 197
Chapter12 MargulisSuperrigidityTheorem 199
§12A. MargulisSuperrigidityTheorem . . . . . . . . . . . . . . . . 199
§12B. MostowRigidityTheorem . . . . . . . . . . . . . . . . . . . . 202
§12C. Whysuperrigidityimpliesarithmeticity. . . . . . . . . . . . 204
§12D. ProofoftheMargulisSuperrigidityTheorem . . . . . . . . 206
§12E. AnA-invariantsection . . . . . . . . . . . . . . . . . . . . . . 210
§12F. Aquicklookatproximality . . . . . . . . . . . . . . . . . . . 212
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
viii CONTENTS
Chapter13 NormalSubgroupsof 217
Γ
§13A. Normalsubgroupsinlatticesofrealrank≥2 . . . . . . . . 217
§13B. Normalsubgroupsinlatticesofrankone . . . . . . . . . . 221
§13C. Γ-equivariantquotientsofG/P (optional) . . . . . . . . . . . 223
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Chapter14 ReductionTheory:AFundamentalSetforG/
Γ
hnotwrittenyeti 231
Chapter15 ArithmeticLatticesinClassicalGroups 233
§15A. ComplexificationofG. . . . . . . . . . . . . . . . . . . . . . . 233
§15B. CalculatingthecomplexificationofG . . . . . . . . . . . . . 235
§15C. Cocompactlatticesinsomeclassicalgroups . . . . . . . . 238
§15D. Isotypicclassicalgroupshaveirreduciblelattices . . . . . 241
§15E. WhatisacentraldivisionalgebraoverF? . . . . . . . . . . 247
§15F. Whatisanabsolutelysimplegroup? . . . . . . . . . . . . . 250
§15G. Absolutelysimpleclassicalgroups. . . . . . . . . . . . . . . 252
§15H. TheLiegroupcorrespondingtoeachF-group. . . . . . . . 254
§15I. Thearithmeticlatticesinclassicalgroups . . . . . . . . . . 255
§15J. WhatarethepossibleHermitianforms? . . . . . . . . . . . 258
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Appendices 269
AppendixA AssumedBackground 269
§A.1. Riemmanianmanifolds . . . . . . . . . . . . . . . . . . . . . . 269
§A.2. Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
§A.3. Liegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
§A.4. Galoistheoryandfieldextensions . . . . . . . . . . . . . . . 276
§A.5. Algebraicnumbersandtranscendentalnumbers . . . . . . 277
§A.6. PolynomialringsandtheNullstellensatz . . . . . . . . . . . 278
§A.7. EisensteinCriterion . . . . . . . . . . . . . . . . . . . . . . . . 281
§A.8. Measuretheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
§A.9. FunctionalAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 282
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
AppendixB WhichClassicalGroupsareIsogenous? 283
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
ix
AppendixC CentralDivisionAlgebrasoverNumberFields 291
§C.1. How to construct central division algebras over number
fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
§C.2. TheBrauergroup . . . . . . . . . . . . . . . . . . . . . . . . . 295
§C.3. Divisionalgebrasarecyclic . . . . . . . . . . . . . . . . . . . 297
§C.4. Simplealgebrasarematrixalgebras . . . . . . . . . . . . . . 298
§C.5. Cohomologicalapproachtodivisionalgebras . . . . . . . . 299
§C.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
