Introduction to Arithmetic Groups Preliminaryversion0.5(April6,2008) [email protected] Dave Witte Morris DepartmentofMathematicsandComputerScience UniversityofLethbridge Lethbridge,Alberta,T1K3M4,Canada [email protected],http://people.uleth.ca/~dave.morris/ April 7, 2008 Copyright(cid:13)c 2001–2008DaveWitteMorris.Allrightsreserved. Permissiontomakecopiesoftheselecturenotesforeducationalorscientificuse, includingmultiplecopiesforclassroomorseminarteaching,isgranted(without fee), provided that any fees charged for the copies are only sufficient to recover the reasonable copying costs, and that all copies include this copyright notice. Specific written permission of the author is required to reproduce or distribute thisbook(inwholeorinpart)forprofitorcommercialadvantage. 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