Arithmetic Groups and Locally Symmetric Spaces Preliminary version — send comments to [email protected] Dave Witte Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078, USA E-mail address: [email protected], http://www.math.okstate.edu/~dwitte Copyright (cid:1)c 2001 Dave Witte. All rights reserved. Permission to make copies of these lecture notes for educational or scientific use, including multiple copies for classroom or seminar teaching, is granted (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 title page and its copyright notice. Specific permission of the author is required to reproduce or distribute this book (in whole or in part) for profit or commercial advantage. List of Chapters (Only 1–6, 8–10, and App. I are available so far) Chapter 1. What is a Locally Symmetric Space? 1 Chapter 2. Geometer’s Introduction to R-rank and Q-rank 13 Chapter 3. Introduction to Semisimple Lie Groups 17 Chapter 4. Some of the Structure of Semisimple Lie Groups 30 Chapter 5. Basic Properties of Lattices 41 Chapter 6. What is an Arithmetic Lattice? 64 Chapter 7. Examples of Lattices 82 Chapter 8. Real Rank 86 Chapter 9. Q-Rank 95 Chapter 10. k-Forms of Classical Groups 108 Chapter 11. Galois Cohomology and k-Forms 134 Chapter 12. Lattices of Extremal Q-rank 135 Chapter 13. Fundamental Domain for G/GZ 139 Chapter 14. Arithmetic Subgroups are Lattices 140 Chapter 15. Zassenhaus Neighborhood 141 Chapter 16. Rigidity, Strong Rigidity and Superrigidity 142 Chapter 17. Amenability vs. Kazhdan’s Property (T) 147 Chapter 18. Ergodic Theory 151 Chapter 19. Proof of the Superrigidity Theorem via Ergodic Theory 155 iii ivPreliminary version (June 9, 2001) List of Chapters (Only 1–6, 8–10, and App. I are available so far) Chapter 20. Proof of the Superrigidity Theorem via Harmonic Maps 158 Chapter 21. Cohomology of Arithmetic Lattices 159 Chapter 22. Normal Subgroups of Real-Rank-One Lattices 161 Chapter 23. Normal Subgroups of Higher-Rank Lattices 162 Chapter 24. Root Systems 170 Chapter 25. Basic Properties of Semisimple Algebraic k-Groups 173 Chapter 26. Actions of Γ on One-Dimensional Spaces 174 Chapter 27. Introduction to Unipotent Dynamics 175 Chapter 28. Other Possible Topics 176 Appendix I. Assumed Background 177 Bibliography 185 Index 188 Contents Chapter 1. What is a Locally Symmetric Space? 1 §1A. Symmetric spaces 1 §1B. How to construct a symmetric space 3 §1C. Locally symmetric spaces 6 §1D. Notes 9 Exercises 9 Chapter 2. Geometer’s Introduction to R-rank and Q-rank 13 §2A. Rank and real rank 13 §2B. Q-rank 14 §2C. Notes 16 Exercises 16 Chapter 3. Introduction to Semisimple Lie Groups 17 §3A. The standing assumptions 17 §3B. Isogenies 17 §3C. What is a semisimple Lie group? 18 §3D. The simple Lie groups 20 §3E. Which classical groups are isogenous? 22 §3F. Notes 25 Exercises 26 Chapter 4. Some of the Structure of Semisimple Lie Groups 30 §4A. G is almost Zariski closed 30 §4B. Real Jordan decomposition 32 §4C. Jacobson-Morosov Lemma 33 §4D. Maximal compact subgroups and the Iwasawa decomposition 33 §4E. Cartan involution and Cartan decomposition 34 §4F. The image of the exponential map 34 v vi Preliminary version (June 9, 2001) Contents §4G. Parabolic subgroups 35 §4H. The normalizer of G 37 §4I. Fundamental group and center of G 38 §4J. Notes 38 Exercises 39 Chapter 5. Basic Properties of Lattices 41 §5A. Definition 41 §5B. Commensurability 43 §5C. Irreducible lattices 44 §5D. Unbounded subsets of Γ\G 45 §5E. Intersection of Γ with other subgroups of G 47 §5F. Borel Density Theorem and some consequences 48 §5G. Proof of the Borel Density Theorem 49 §5H. Γ is finitely presented 51 §5I. Γ has a torsion-free subgroup of finite index 53 §5J. Γ has a nonabelian free subgroup 56 §5K. Notes 59 Exercises 60 Chapter 6. What is an Arithmetic Lattice? 64 §6A. Definition of arithmetic lattices 64 §6B. Margulis Arithmeticity Theorem 67 §6C. Commensurability criterion for arithmeticity 68 §6D. Why superrigidity implies arithmeticity 69 §6E. Unipotent elements of GZ: the Godement Compactness Criterion 70 §6F. How to make an arithmetic lattice 72 §6G. Restriction of scalars 73 §6H. Notes 78 Exercises 79 Chapter 7. Examples of Lattices 82 §7A. Cocompact arithmetic lattices in SL(2,R) 82 §7B. Teichmu¨ller space and moduli space of lattices in SL(2,R) 82 §7C. Noncocompact lattices in SL(3,R) 82 §7D. Arithmetic lattices in SO(1,n) 85 §7E. Some non-arithmetic lattices in SO(1,n) 85 §7F. Notes 85 Chapter 8. Real Rank 86 §8A. R-split tori 86 §8B. Definition of real rank 87 §8C. Relation to geometry 88 Preliminary version (June 9, 2001) Contents vii §8D. Parabolic subgroups 89 §8E. Groups of real rank zero 89 §8F. Groups of real rank one 90 §8G. Groups of higher real rank 91 §8H. Notes 93 Exercises 94 Chapter 9. Q-Rank 95 §9A. Q-split tori 95 §9B. Q-rank of an arithmetic lattice 96 §9C. Isogenies over Q 97 §9D. Q-rank of any lattice 98 §9E. The possible Q-ranks 99 §9F. Lattices of Q-rank zero 100 §9G. Lattices of Q-rank one 101 §9H. Lattices of higher Q-rank 102 §9I. Parabolic Q-subgroups 103 §9J. The large-scale geometry of Γ\X 104 §9K. Notes 107 Exercises 107 Chapter 10. k-Forms of Classical Groups 108 §10A. Complexification of G 108 §10B. Calculating the complexification of G 110 §10C. Cocompact lattices in some classical groups 112 §10D. Isotypic classical groups have irreducible lattices 114 §10E. What is an absolutely simple group? 119 §10F. Absolutely simple classical groups 119 §10G. The arithmetic lattices in classical groups 123 §10H. Very basic algebraic number theory 123 §10I. Central division algebras over number fields 125 §10J. Notes 130 Exercises 131 Chapter 11. Galois Cohomology and k-Forms 134 §11A. The Tits Classification 134 §11B. Inner forms and outer forms 134 §11C. Quasi-split groups 134 Chapter 12. Lattices of Extremal Q-rank 135 §12A. Construction using Galois cohomology 135 §12B. Explicit construction of the Lie algebra 136 §12C. Notes 138 viii Preliminary version (June 9, 2001) Contents Chapter 13. Fundamental Domain for G/GZ 139 §13A. Godement Criterion for compactness of H/HZ 139 §13B. Dirichlet’s Unit Theorem 139 §13C. Reduction theory: a weak fundamental domain for G/GZ 139 §13D. Large-scale geometry of Γ\X 139 Chapter 14. Arithmetic Subgroups are Lattices 140 §14A. G/GZ has finite volume 140 §14B. Divergent sequences in G/GZ 140 §14C. Proof of Godement’s Criterion 140 Chapter 15. Zassenhaus Neighborhood 141 §15A. Zassenhaus neighborhood 141 §15B. Lower bound on vol(Γ\G) 141 §15C. Existence of maximal lattices 141 §15D. Noncocompact lattices have unipotent elements 141 §15E. Weak fundamental domain for G/Γ 141 Chapter 16. Rigidity, Strong Rigidity and Superrigidity 142 §16A. Deformations of Γ 142 §16B. Representation Varieties 142 §16C. Mostow Rigidity Theorem 142 §16D. Quasi-isometry rigidity 143 §16E. Margulis Superrigidity Theorem 144 §16F. Geometric superrigidity 146 §16G. A nonarithmetic superrigid group 146 §16H. Notes 146 Chapter 17. Amenability vs. Kazhdan’s Property (T) 147 §17A. Amenability 147 §17B. Kazhdan’s property (T) 148 §17C. Notes 150 Chapter 18. Ergodic Theory 151 §18A. Tameness vs. ergodicity 151 §18B. Mean Ergodic Theorem 151 §18C. Pointwise Ergodic Theorem 151 §18D. Multiplicative Ergodic Theorem 151 §18E. Moore Ergodicity Theorem 151 §18F. Cocycle Superrigidity Theorem 154 Chapter 19. Proof of the Superrigidity Theorem via Ergodic Theory 155 §19A. The Zariski closure of ρ(Γ) 155 §19B. Measurable equivariant map on the boundary 155 Preliminary version (June 9, 2001) Contents ix §19C. Action on the space of probability measures 156 §19D. Stabilizers of probability measures 156 §19E. Equivariant maps are rational 156 §19F. Notes 157 Chapter 20. Proof of the Superrigidity Theorem via Harmonic Maps 158 Chapter 21. Cohomology of Arithmetic Lattices 159 §21A. Cohomology 159 §21B. Hodge theory and deRham cohomology 159 §21C. Cohomology vanishing for higher-rank lattices 159 §21D. Hermitian symmetric spaces and infinite fundamental groups 159 §21E. Lattices in semisimple groups with infinite center 159 §21F. Bounded cohomology 160 §21G. Notes 160 Chapter 22. Normal Subgroups of Real-Rank-One Lattices 161 §22A. Quotients of hyperbolic groups 161 §22B. Abelian quotients of lattices in SO(1,n) 161 Chapter 23. Normal Subgroups of Higher-Rank Lattices 162 §23A. Γ is almost simple 162 §23B. Unipotent generators 163 §23C. The Congruence Subgroup Property 166 §23D. Bounded generation 166 §23E. Lattices with no torsion-free subgroup of finite index 168 §23F. Almost-normal subgroups 168 §23G. Conjugation-invariant subsemigroups 168 §23H. Notes 168 Exercises 168 Chapter 24. Root Systems 170 §24A. Roots of complex Lie algebras 170 §24B. Definition of roots 170 §24C. Classification of semisimple Lie algebras over C 171 §24D. Dynkin diagrams 171 §24E. The root system BC 172 n §24F. Real roots 172 §24G. Q-roots 172 §24H. Notes 172 Chapter 25. Basic Properties of Semisimple Algebraic k-Groups 173 §25A. Kneser-Tits Conjecture 173 §25B. BN-pairs 173 x Preliminary version (June 9, 2001) Contents §25C. Tits building 173 §25D. Normal subgroups of GQ 173 §25E. Connected unipotent subgroups are Zariski closed 173 Chapter 26. Actions of Γ on One-Dimensional Spaces 174 §26A. Isometric actions on trees 174 §26B. Continuous actions on the circle 174 §26C. Smooth actions on the circle 174 §26D. Smooth actions of Kazhdan groups on the circle 174 §26E. Higher dimensions 174 §26F. Notes 174 Chapter 27. Introduction to Unipotent Dynamics 175 §27A. Ratner’s Theorem 175 §27B. Oppenheim Conjecture 175 §27C. The return of unipotent orbits 175 Chapter 28. Other Possible Topics 176 §28A. Automorphic forms 176 §28B. Orbifolds 176 §28C. Patterson-Sullivan Measure 176 §28D. Compactifications of X and Γ\X 176 Appendix I. Assumed Background 177 §I.A. Riemmanian manifolds 177 §I.B. Geodesics 177 §I.C. Lie groups 178 §I.D. Galois theory and field extensions 181 §I.E. Polynomial rings and the Nullstellensatz 182 §I.F. Notes 183 Bibliography 185 Index 188