Introduction to Arithmetic Groups Dave Witte Morris

232 Pages·2003·2.18 MB·English

by  Dave Witte

Introduction to Arithmetic Groups
Preliminary version (February 27, 2003)
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Dave Witte Morris
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, 2002, 2003 Dave Witte Morris. All rights reserved.
Permission to make copies of these lecture notes for educational or scientific use, including multiple copies
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Acknowledgments
In writing this book, I have benefited from the comments and suggestions of many colleagues,
including Marc Burger, Indira Chatterjee, 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 E´. Ghys, D. Gaboriau and the other members of the E´cole Normale
Sup´erieure de Lyon for the stimulating conversations and invitation to speak that were the impetus
for this work, and B. Farb, A. Eskin and the other members of the University of Chicago math-
ematics department for encouraging me to write this introduction, for the opportunity to 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 partially
supported by a research grant from the National Science Foundation (DMS–0100438).
iii
List of Chapters
(Only 1–11, 25, 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 32
Chapter 5. Basic Properties of Lattices 44
Chapter 6. What is an Arithmetic Lattice? 68
Chapter 7. Examples of Lattices 88
Chapter 8. Real Rank 112
Chapter 9. Q-Rank 121
Chapter 10. Arithmetic Lattices in Classical Groups 135
Chapter 11. Central division algebras over number fields 164
Chapter 12. Galois Cohomology and Q-Forms 173
Chapter 13. Lattices of Extremal Q-rank 175
Chapter 14. Fundamental Domain for G/GZ 179
Chapter 15. Arithmetic Subgroups are Lattices 181
Chapter 16. Zassenhaus Neighborhood 182
Chapter 17. Rigidity, Strong Rigidity and Superrigidity 183
Chapter 18. Root Systems 189
iv
Preliminary version (February 27, 2003) List of Chapters(Only 1–11, 25, and App. I are available so farv)
Chapter 19. Basic Properties of Semisimple Algebraic k-Groups 193
Chapter 20. Amenability vs. Kazhdan’s Property (T) 194
Chapter 21. Introduction to Ergodic Theory 200
Chapter 22. Proof of the Margulis Superrigidity Theorem 204
Chapter 23. Normal Subgroups of Higher-Rank Lattices 207
Chapter 24. Cohomology of Arithmetic Lattices 216
Chapter 25. Actions on the Circle 218
Appendix I. Assumed Background 250
List of Notation 262
Index 263
Contents
Acknowledgments iii
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
§1E. 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
§2D. 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
§3G. References 28
Exercises 28
Chapter 4. Some of the Structure of Semisimple Lie Groups 32
§4A. G is almost Zariski closed 32
§4B. Real Jordan decomposition 34
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Preliminary version (February 27, 2003) Contents vii
§4C. Jacobson-Morosov Lemma 35
§4D. Maximal compact subgroups and the Iwasawa decomposition 35
§4E. Cartan involution and Cartan decomposition 36
§4F. The image of the exponential map 36
§4G. Parabolic subgroups 37
§4H. The normalizer of G 39
§4I. Fundamental group and center of G 40
§4J. Notes 40
§4K. References 40
Exercises 41
Chapter 5. Basic Properties of Lattices 44
§5A. Definition 44
§5B. Commensurability 46
§5C. Irreducible lattices 47
§5D. Unbounded subsets of Γ\G 48
§5E. Intersection of Γ with other subgroups of G 50
§5F. Borel Density Theorem and some consequences 51
§5G. Proof of the Borel Density Theorem 52
§5H. Γ is finitely presented 54
§5I. Γ has a torsion-free subgroup of finite index 56
§5J. Γ has a nonabelian free subgroup 59
§5K. Notes 62
§5L. References 63
Exercises 63
Chapter 6. What is an Arithmetic Lattice? 68
§6A. Definition of arithmetic lattices 68
§6B. Margulis Arithmeticity Theorem 71
§6C. Commensurability criterion for arithmeticity 73
§6D. Why superrigidity implies arithmeticity 73
§6E. Unipotent elements of GZ: the Godement Compactness Criterion 75
§6F. How to make an arithmetic lattice 76
§6G. Restriction of scalars 78
§6H. Notes 83
§6I. References 83
Exercises 84
Chapter 7. Examples of Lattices 88
§7A. Arithmetic lattices in SL(2,R) 88
§7B. Teichmu¨ller space and moduli space of lattices in SL(2,R) 91
§7C. Arithmetic lattices in SO(1,n) 92
viii Preliminary version (February 27, 2003) Contents
§7D. Some nonarithmetic lattices in SO(1,n) 94
§7E. Noncocompact lattices in SL(3,R) 101
§7F. Cocompact lattices in SL(3,R) 103
§7G. Lattices in SL(n,R) 106
§7H. Quaternion algebras over a field F 107
§7I. Notes 108
§7J. References 108
Exercises 108
Chapter 8. Real Rank 112
§8A. R-split tori 112
§8B. Definition of real rank 113
§8C. Relation to geometry 114
§8D. Parabolic subgroups 115
§8E. Groups of real rank zero 115
§8F. Groups of real rank one 116
§8G. Groups of higher real rank 118
§8H. Notes 120
§8I. References 120
Exercises 120
Chapter 9. Q-Rank 121
§9A. Q-split tori 121
§9B. Q-rank of an arithmetic lattice 122
§9C. Isogenies over Q 124
§9D. Q-rank of any lattice 125
§9E. The possible Q-ranks 125
§9F. Lattices of Q-rank zero 126
§9G. Lattices of Q-rank one 128
§9H. Lattices of higher Q-rank 128
§9I. Parabolic Q-subgroups 129
§9J. The large-scale geometry of Γ\X 131
§9K. Notes 133
§9L. References 133
Exercises 134
Chapter 10. Arithmetic Lattices in Classical Groups 135
§10A. Complexification of G 135
§10B. Calculating the complexification of G 137
§10C. Cocompact lattices in some classical groups 139
§10D. Isotypic classical groups have irreducible lattices 141
§10E. What is a central division algebra over F? 146
Preliminary version (February 27, 2003) Contents ix
§10F. What is an absolutely simple group? 148
§10G. Absolutely simple classical groups 149
§10H. The Lie group corresponding to each F-group 151
§10I. The arithmetic lattices in classical groups 152
§10J. What are the possible Hermitian forms? 154
§10K. Notes 160
§10L. References 160
Exercises 160
Chapter 11. Central division algebras over number fields 164
§11A. How to construct central division algebras over number fields 164
§11B. The Brauer group 167
§11C. Division algebras are cyclic 169
§11D. Simple algebras are matrix algebras 170
§11E. Cohomological approach to division algebras 171
§11F. Notes 172
§11G. References 172
Exercises 172
Chapter 12. Galois Cohomology and Q-Forms 173
§12A. Using Galois Cohomology to find the F-forms of classical groups 173
§12B. The Tits Classification 174
§12C. Inner forms and outer forms 174
§12D. Quasi-split groups 174
§12E. References 174
Exercises 174
Chapter 13. Lattices of Extremal Q-rank 175
§13A. Construction using Galois cohomology 175
§13B. Explicit construction of the Lie algebra 176
§13C. Notes 178
§13D. References 178
Exercises 178
Chapter 14. Fundamental Domain for G/GZ 179
§14A. Godement Criterion for compactness of H/HZ 179
§14B. Dirichlet’s Unit Theorem 179
§14C. Reduction theory: a weak fundamental domain for G/GZ 179
§14D. Large-scale geometry of Γ\X 180
§14E. References 180
Exercises 180
Chapter 15. Arithmetic Subgroups are Lattices 181
x Preliminary version (February 27, 2003) Contents
§15A. G/GZ has finite volume 181
§15B. Divergent sequences in G/GZ 181
§15C. Proof of Godement’s Criterion 181
§15D. References 181
Exercises 181
Chapter 16. Zassenhaus Neighborhood 182
§16A. Zassenhaus neighborhood 182
§16B. Lower bound on vol(Γ\G) 182
§16C. Existence of maximal lattices 182
§16D. Noncocompact lattices have unipotent elements 182
§16E. Weak fundamental domain for G/Γ 182
§16F. References 182
Exercises 182
Chapter 17. Rigidity, Strong Rigidity and Superrigidity 183
§17A. Deformations of Γ 183
§17B. Representation Varieties 183
§17C. Mostow Rigidity Theorem 183
§17D. Proof of the Mostow Rigidity Theorem 184
§17E. Quasi-isometry rigidity 184
§17F. Margulis Superrigidity Theorem 185
§17G. Homomorphisms into compact groups 187
§17H. Geometric superrigidity 187
§17I. A nonarithmetic superrigid group 187
§17J. Notes 187
§17K. References 188
Chapter 18. Root Systems 189
§18A. Roots of complex Lie algebras 189
§18B. Definition of roots 189
§18C. Classification of semisimple Lie algebras over C 190
§18D. Dynkin diagrams 190
§18E. The root system BC 191
n
§18F. Real roots 191
§18G. Q-roots 191
§18H. Notes 192
§18I. References 192
Exercises 192
Chapter 19. Basic Properties of Semisimple Algebraic k-Groups 193
§19A. Kneser-Tits Conjecture 193
§19B. BN-pairs 193