Introduction to Arithmetic Groups Dave Witte Morris

266 Pages·2006·1.99 MB·English

Introduction to Arithmetic Groups
Preliminary version 0.1 (July 14, 2006)
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Dave Witte Morris
Department of Mathematics and Computer Science
University of Lethbridge
Lethbridge, Alberta, T1K 3M4, Canada
E-mail address: [email protected], http://people.uleth.ca/~dave.morris/
Copyright (cid:13)c 2001–2005 Dave Witte Morris. All rights reserved.
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Acknowledgments
Inwritingthisbook,Ihavebenefitedfromthecommentsandsuggestionsofmanycolleagues,
including Marc Burger, Indira Chatterji, Alessandra Iozzi, Anders Karlsson, Sean Keel, Nicolas
Monod, Hee Oh, Yehuda Shalom, Shmuel Weinberger, Barak Weiss, and Kevin Whyte.
I have also benefited greatly from discussions with Scot Adams, S.G. Dani, Benson Farb,
G.A. Margulis, Gopal Prasad, M.S. Raghunathan, T. N. Venkataramana, and Robert J. Zimmer.
They (and others) may note that some of the comments they made to me over the years have
been reproduced here almost verbatim.
I would like to thank É. Ghys, D. Gaboriau and the other members of the École Normale
Supérieure de Lyon for the stimulating conversations and invitation to speak that were the
impetusforthiswork,andB.Farb,A.EskinandtheothermembersoftheUniversityofChicago
mathematicsdepartmentforencouragingmetowritethisintroduction,fortheopportunityto
lecture from it, and for providing a stimulating audience.
I am grateful to ENS–Lyon, the University of Bielefeld, the Isaac Newton Institute for Math-
ematical Sciences, and the Tata Institute for Fundamental Research for their warm hospitality
while various of the chapters were being written. The preparation of this manuscript was
partiallysupportedbyaresearchgrantfromtheNationalScienceFoundation(DMS–0100438).
iii
List of Chapters
(Only 1–11, 27, and App. I are available so far)
Acknowledgments iii
Chapter 1. What is a Locally Symmetric Space? 1
Chapter 2. Geometer’s Introduction to R-rank and Q-rank 14
Chapter 3. Introduction to Semisimple Lie Groups 19
Chapter 4. Some of the Structure of Semisimple Lie Groups 33
Chapter 5. Basic Properties of Lattices 45
Chapter 6. What is an Arithmetic Lattice? 70
Chapter 7. Examples of Lattices 91
Chapter 8. Real Rank 116
Chapter 9. Q-Rank 126
Chapter 10. Arithmetic Lattices in Classical Groups 141
Chapter 11. Central Division Algebras over Number Fields 171
Chapter 12. Galois Cohomology and Q-Forms 182
Chapter 13. Lattices of Extremal Q-rank 188
Chapter 14. Fundamental Domain for G/GZ 192
Chapter 15. Arithmetic Subgroups are Lattices 195
Chapter 16. Zassenhaus Neighborhood 196
Chapter 17. Rigidity, Strong Rigidity and Superrigidity 197
Chapter 18. Root Systems 203
iv
Preliminary version (July 14, 2006) List of Chapters (Only1–11,27,andApp.Iareavailablesofar)v
Chapter 19. Basic Properties of Semisimple Algebraic k-Groups 207
Chapter 20. Amenability vs. Kazhdan’s Property (T) 208
Chapter 21. Introduction to Ergodic Theory 214
Chapter 22. Proof of the Margulis Superrigidity Theorem 219
Chapter 23. Normal Subgroups of Higher-Rank Lattices 222
Chapter 24. Cohomology of Arithmetic Lattices 229
Chapter 25. Bounded Generation 231
Chapter 26. The Congruence Subgroup Property 254
Chapter 27. Actions on the Circle 256
Appendix I. Assumed Background 289
List of Notation 301
Index 302
Contents
Acknowledgments iii
Chapter 1. What is a Locally Symmetric Space? 1
§1A. Symmetric spaces 1
§1B. How to construct a symmetric space 4
§1C. Locally symmetric spaces 6
§1D. Notes 10
References 10
Exercises 10
Chapter 2. Geometer’s Introduction to R-rank and Q-rank 14
§2A. Rank and real rank 14
§2B. Q-rank 15
§2C. Notes 18
References 18
Exercises 18
Chapter 3. Introduction to Semisimple Lie Groups 19
§3A. The standing assumptions 19
§3B. Isogenies 19
§3C. What is a semisimple Lie group? 20
§3D. The simple Lie groups 22
§3E. Which classical groups are isogenous? 24
§3F. Notes 28
References 28
Exercises 28
Chapter 4. Some of the Structure of Semisimple Lie Groups 33
§4A. G is almost Zariski closed 33
§4B. Real Jordan decomposition 35
vi
Preliminary version (July 14, 2006) Contents vii
§4C. Jacobson-Morosov Lemma 36
§4D. Maximal compact subgroups and the Iwasawa decomposition 37
§4E. Cartan involution and Cartan decomposition 37
§4F. The image of the exponential map 37
§4G. Parabolic subgroups 38
§4H. The normalizer of G 41
§4I. Fundamental group and center of G 41
§4J. Notes 42
References 42
Exercises 43
Chapter 5. Basic Properties of Lattices 45
§5A. Definition 45
§5B. Commensurability 47
§5C. Irreducible lattices 48
§5D. Unbounded subsets of \G 49
Γ
§5E. Intersection of with other subgroups of G 52
Γ
§5F. Borel Density Theorem and some consequences 52
§5G. Proof of the Borel Density Theorem 54
§5H. is finitely presented 55
Γ
§5I. has a torsion-free subgroup of finite index 58
Γ
§5J. has a nonabelian free subgroup 61
Γ
§5K. Notes 64
References 65
Exercises 65
Chapter 6. What is an Arithmetic Lattice? 70
§6A. Definition of arithmetic lattices 70
§6B. Margulis Arithmeticity Theorem 74
§6C. Commensurability criterion for arithmeticity 75
§6D. Why superrigidity implies arithmeticity 75
§6E. Unipotent elements of GZ: the Godement Compactness Criterion 77
§6F. How to make an arithmetic lattice 78
§6G. Restriction of scalars 80
§6H. Notes 85
References 86
Exercises 86
Chapter 7. Examples of Lattices 91
§7A. Arithmetic lattices in SL(2,R) 91
§7B. Teichmüller space and moduli space of lattices in SL(2,R) 95
§7C. Arithmetic lattices in SO(1,n) 95
viii Preliminary version (July 14, 2006) Contents
§7D. Some nonarithmetic lattices in SO(1,n) 98
§7E. Noncocompact lattices in SL(3,R) 104
§7F. Cocompact lattices in SL(3,R) 107
§7G. Lattices in SL(n,R) 110
§7H. Quaternion algebras over a field F 110
§7I. Notes 111
References 112
Exercises 112
Chapter 8. Real Rank 116
§8A. R-split tori 116
§8B. Definition of real rank 117
§8C. Relation to geometry 119
§8D. Parabolic subgroups 119
§8E. Groups of real rank zero 119
§8F. Groups of real rank one 120
§8G. Groups of higher real rank 122
§8H. Notes 124
References 124
Exercises 124
Chapter 9. Q-Rank 126
§9A. Q-split tori 126
§9B. Q-rank of an arithmetic lattice 127
§9C. Isogenies over Q 129
§9D. Q-rank of any lattice 130
§9E. The possible Q-ranks 131
§9F. Lattices of Q-rank zero 132
§9G. Lattices of Q-rank one 133
§9H. Lattices of higher Q-rank 134
§9I. Parabolic Q-subgroups 135
§9J. The large-scale geometry of \X 136
Γ
§9K. Notes 139
References 139
Exercises 139
Chapter 10. Arithmetic Lattices in Classical Groups 141
§10A. Complexification of G 141
§10B. Calculating the complexification of G 143
§10C. Cocompact lattices in some classical groups 145
§10D. Isotypic classical groups have irreducible lattices 147
§10E. What is a central division algebra over F? 152
Preliminary version (July 14, 2006) Contents ix
§10F. What is an absolutely simple group? 155
§10G. Absolutely simple classical groups 155
§10H. The Lie group corresponding to each F-group 157
§10I. The arithmetic lattices in classical groups 158
§10J. What are the possible Hermitian forms? 161
§10K. Notes 166
References 167
Exercises 167
Chapter 11. Central Division Algebras over Number Fields 171
§11A. How to construct central division algebras over number fields 171
§11B. The Brauer group 175
§11C. Division algebras are cyclic 176
§11D. Simple algebras are matrix algebras 177
§11E. Cohomological approach to division algebras 178
§11F. Notes 179
References 179
Exercises 179
Chapter 12. Galois Cohomology and Q-Forms 182
§12A. Introduction to Galois cohomology 182
§12B. The Borel-Harder Theorem and how to use it 185
§12C. Using Galois cohomology to find the F-forms of classical groups 186
§12D. The Tits Classification 187
§12E. Inner forms and outer forms 187
§12F. Quasi-split groups 187
References 187
Exercises 187
Chapter 13. Lattices of Extremal Q-rank 188
§13A. Construction using Galois cohomology 188
§13B. Explicit construction of the Lie algebra 189
§13C. Notes 191
References 191
Exercises 191
Chapter 14. Fundamental Domain for G/GZ 192
§14A. Godement Criterion for compactness of H/HZ 192
§14B. Dirichlet’s Unit Theorem 192
§14C. Reduction theory: a weak fundamental domain for G/GZ 193
§14D. Large-scale geometry of \X 193
Γ
References 193
Exercises 193
x Preliminary version (July 14, 2006) Contents
Chapter 15. Arithmetic Subgroups are Lattices 195
§15A. G/GZ has finite volume 195
§15B. Divergent sequences in G/GZ 195
§15C. Proof of Godement’s Criterion 195
References 195
Exercises 195
Chapter 16. Zassenhaus Neighborhood 196
§16A. Zassenhaus neighborhood 196
§16B. Lower bound on vol( \G) 196
Γ
§16C. Existence of maximal lattices 196
§16D. Noncocompact lattices have unipotent elements 196
§16E. Weak fundamental domain for G/ 196
Γ
References 196
Exercises 196
Chapter 17. Rigidity, Strong Rigidity and Superrigidity 197
§17A. Deformations of 197
Γ
§17B. Representation Varieties 197
§17C. Mostow Rigidity Theorem 197
§17D. Proof of the Mostow Rigidity Theorem 198
§17E. Quasi-isometry rigidity 198
§17F. Margulis Superrigidity Theorem 199
§17G. Homomorphisms into compact groups 201
§17H. Geometric superrigidity 201
§17I. A nonarithmetic superrigid group 201
§17J. Notes 202
References 202
Chapter 18. Root Systems 203
§18A. Roots of complex Lie algebras 203
§18B. Definition of roots 203
§18C. Classification of semisimple Lie algebras over C 204
§18D. Dynkin diagrams 204
§18E. The root system BC 205
n
§18F. Real roots 205
§18G. Q-roots 205
§18H. Notes 206
References 206
Exercises 206
Chapter 19. Basic Properties of Semisimple Algebraic k-Groups 207
§19A. Kneser-Tits Conjecture 207