Introduction to Arithmetic Groups Dave Witte

196 Pages·2001·2.09 MB·English

Introduction to Arithmetic Groups
Preliminary version (October 23, 2001)
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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 written 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 1–12 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 18
Chapter 4. Some of the Structure of Semisimple Lie Groups 31
Chapter 5. Basic Properties of Lattices 42
Chapter 6. What is an Arithmetic Lattice? 65
Chapter 7. Division Algebras and Hermitian Forms 84
Chapter 8. Central division algebras over number fields 91
Chapter 9. Examples of Lattices 101
Chapter 10. Real Rank 124
Chapter 11. Q-Rank 133
Chapter 12. Arithmetic Lattices in Classical Groups 147
Chapter 13. Galois Cohomology and k-Forms 167
Chapter 14. Lattices of Extremal Q-rank 169
Chapter 15. Fundamental Domain for G/GZ 173
Chapter 16. Arithmetic Subgroups are Lattices 175
Chapter 17. Zassenhaus Neighborhood 176
Chapter 18. Root Systems 177
Chapter 19. Basic Properties of Semisimple Algebraic k-Groups 180
iii
ivPreliminary version (October 23, 2001) List of Chapters (Only 1–12 and App. I are available so far)
Chapter 20. Amenability vs. Kazhdan’s Property (T) 181
Appendix I. Assumed Background 185
Appendix II. Some Results From Volume Two 195
Bibliography 199
List of Notation 202
Index 203
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 18
§3A. The standing assumptions 18
§3B. Isogenies 18
§3C. What is a semisimple Lie group? 19
§3D. The simple Lie groups 21
§3E. Which classical groups are isogenous? 23
§3F. Notes 26
Exercises 27
Chapter 4. Some of the Structure of Semisimple Lie Groups 31
§4A. G is almost Zariski closed 31
§4B. Real Jordan decomposition 33
§4C. Jacobson-Morosov Lemma 34
§4D. Maximal compact subgroups and the Iwasawa decomposition 34
§4E. Cartan involution and Cartan decomposition 35
§4F. The image of the exponential map 35
v
vi Preliminary version (October 23, 2001) Contents
§4G. Parabolic subgroups 36
§4H. The normalizer of G 38
§4I. Fundamental group and center of G 39
§4J. Notes 40
Exercises 40
Chapter 5. Basic Properties of Lattices 42
§5A. Definition 42
§5B. Commensurability 44
§5C. Irreducible lattices 45
§5D. Unbounded subsets of Γ\G 46
§5E. Intersection of Γ with other subgroups of G 48
§5F. Borel Density Theorem and some consequences 49
§5G. Proof of the Borel Density Theorem 50
§5H. Γ is finitely presented 52
§5I. Γ has a torsion-free subgroup of finite index 54
§5J. Γ has a nonabelian free subgroup 57
§5K. Notes 60
Exercises 61
Chapter 6. What is an Arithmetic Lattice? 65
§6A. Definition of arithmetic lattices 65
§6B. Margulis Arithmeticity Theorem 68
§6C. Commensurability criterion for arithmeticity 70
§6D. Why superrigidity implies arithmeticity 70
§6E. Unipotent elements of GZ: the Godement Compactness Criterion 72
§6F. How to make an arithmetic lattice 73
§6G. Restriction of scalars 75
§6H. Notes 80
Exercises 80
Chapter 7. Division Algebras and Hermitian Forms 84
§7A. Quaternion algebras over F 84
§7B. What is a central division algebra over F? 85
§7C. Symplectic forms 87
§7D. Symmetric, bilinear forms 88
§7E. Hermitian forms 89
§7F. Notes 90
Exercises 90
Chapter 8. Central division algebras over number fields 91
§8A. How to construct central division algebras over number fields 91
§8B. The Brauer group 94
Preliminary version (October 23, 2001) Contents vii
§8C. Division algebras are cyclic 95
§8D. Simple algebras are matrix algebras 96
§8E. Cohomological approach to division algebras 98
§8F. Notes 99
Exercises 99
Chapter 9. Examples of Lattices 101
§9A. Arithmetic lattices in SL(2,R) 101
§9B. Teichmu¨ller space and moduli space of lattices in SL(2,R) 104
§9C. Arithmetic lattices in SO(1,n) 105
§9D. Some nonarithmetic lattices in SO(1,n) 107
§9E. Noncocompact lattices in SL(3,R) 114
§9F. Cocompact lattices in SL(3,R) 116
§9G. Lattices in SL(n,R) 119
§9H. Notes 120
Exercises 120
Chapter 10. Real Rank 124
§10A. R-split tori 124
§10B. Definition of real rank 125
§10C. Relation to geometry 126
§10D. Parabolic subgroups 127
§10E. Groups of real rank zero 127
§10F. Groups of real rank one 128
§10G. Groups of higher real rank 129
§10H. Notes 131
Exercises 132
Chapter 11. Q-Rank 133
§11A. Q-split tori 133
§11B. Q-rank of an arithmetic lattice 134
§11C. Isogenies over Q 135
§11D. Q-rank of any lattice 137
§11E. The possible Q-ranks 137
§11F. Lattices of Q-rank zero 138
§11G. Lattices of Q-rank one 140
§11H. Lattices of higher Q-rank 140
§11I. Parabolic Q-subgroups 141
§11J. The large-scale geometry of Γ\X 143
§11K. Notes 145
Exercises 145
Chapter 12. Arithmetic Lattices in Classical Groups 147
viii Preliminary version (October 23, 2001) Contents
§12A. Complexification of G 147
§12B. Calculating the complexification of G 149
§12C. Cocompact lattices in some classical groups 151
§12D. Isotypic classical groups have irreducible lattices 153
§12E. What is an absolutely simple group? 158
§12F. Absolutely simple classical groups 159
§12G. The Lie group corresponding to each F-group 160
§12H. The arithmetic lattices in classical groups 161
§12I. Notes 164
Exercises 164
Chapter 13. Galois Cohomology and k-Forms 167
§13A. Using Galois Cohomology to find the F-forms of classical groups 167
§13B. The Tits Classification 168
§13C. Inner forms and outer forms 168
§13D. Quasi-split groups 168
Chapter 14. Lattices of Extremal Q-rank 169
§14A. Construction using Galois cohomology 169
§14B. Explicit construction of the Lie algebra 170
§14C. Notes 172
Chapter 15. Fundamental Domain for G/GZ 173
§15A. Godement Criterion for compactness of H/HZ 173
§15B. Dirichlet’s Unit Theorem 173
§15C. Reduction theory: a weak fundamental domain for G/GZ 173
§15D. Large-scale geometry of Γ\X 174
§15E. Exercises 174
Chapter 16. Arithmetic Subgroups are Lattices 175
§16A. G/GZ has finite volume 175
§16B. Divergent sequences in G/GZ 175
§16C. Proof of Godement’s Criterion 175
Chapter 17. Zassenhaus Neighborhood 176
§17A. Zassenhaus neighborhood 176
§17B. Lower bound on vol(Γ\G) 176
§17C. Existence of maximal lattices 176
§17D. Noncocompact lattices have unipotent elements 176
§17E. Weak fundamental domain for G/Γ 176
Chapter 18. Root Systems 177
§18A. Roots of complex Lie algebras 177
§18B. Definition of roots 177
Preliminary version (October 23, 2001) Contents ix
§18C. Classification of semisimple Lie algebras over C 178
§18D. Dynkin diagrams 178
§18E. The root system BC 179
n
§18F. Real roots 179
§18G. Q-roots 179
§18H. Notes 179
Chapter 19. Basic Properties of Semisimple Algebraic k-Groups 180
§19A. Kneser-Tits Conjecture 180
§19B. BN-pairs 180
§19C. Tits building 180
§19D. Normal subgroups of GQ 180
§19E. Connected unipotent subgroups are Zariski closed 180
Chapter 20. Amenability vs. Kazhdan’s Property (T) 181
§20A. Amenability 181
§20B. Kazhdan’s property (T) 182
§20C. Notes 184
Appendix I. Assumed Background 185
§I.A. Riemmanian manifolds 185
§I.B. Geodesics 185
§I.C. Lie groups 186
§I.D. Galois theory and field extensions 189
§I.E. Algebraic numbers and transcendental numbers 190
§I.F. Polynomial rings and the Nullstellensatz 191
§I.G. Eisenstein Criterion 193
§I.H. Notes 193
Exercises 193
Appendix II. Some Results From Volume Two 195
§II.A. Γ is almost simple 195
§II.B. The Congruence Subgroup Property 195
§II.C. Lattices with no torsion-free subgroup of finite index 196
§II.D. Margulis Superrigidity Theorem 196
Bibliography 199
List of Notation 202
Index 203
Chapter 1
What is a Locally Symmetric Space?
In this chapter, we give a geometric introduction to the notion of a symmetric space or a locally
symmetric space, and explain the central role played by simple Lie groups and their lattice sub-
groups. This material is not a prerequisite for reading any of the later chapters, except Chap. 2;
it is intended to provide a geometric motivation for the study of lattices in semisimple Lie groups,
which is the main topic of the rest of the book.
1A. Symmetric spaces
The nicest Riemannian manifolds are homogeneous. This means that every point looks exactly like
every other point.
(1.1) Definition. ARiemannianmanifoldX isahomogeneous space ifitsisometrygroupIsom(X)
acts transitively. That is, for every x,y ∈ X, there is an isometry φ of X, such that φ(x) = y.
(1.2) Example. Here are some elementary examples of (simply connected) homogeneous spaces.
• The round sphere Sn = {x ∈ Rn+1 | (cid:8)x(cid:8) = 1}. Rotations are the only orientation-
preserving isometries of Sn, so Isom(Sn)◦ = SO(n+1). Any point on Sn can be rotated
to any other point, so Sn is homogeneous.
• Euclidean space Rn. Any orientation-preserving isometry of Rn is a combination of a
translation and a rotation, so Isom(Rn)◦ = SO(n)(cid:4)Rn. Any point in Rn can be translated
to any other point, so Rn is homogeneous.
• The hyperbolic plane H2 = {z ∈ C | Imz > 0}, where the inner product on T Hn is given
z
by
1
(cid:10)u | v(cid:11) = (cid:10)u | v(cid:11) .
H2 4(Imz)2 R2
OnemayshowthatIsom(H2)◦ isisomorphictoPSL(2,R)◦ = SL(2,R)/{±1},bynotingthat
SL(2,R) acts on H2 by linear-fractional transformations z (cid:13)→ (az +b)/(cz +d), and con-
firming, by calculation, that these linear-fractional transformations preserve the hyperbolic
metric.
• Hyperbolic space Hn = {x ∈ Rn | x > 0}, where the inner product on T Hn is given by
n x
1
(cid:10)u | v(cid:11)Hn = 4x2(cid:10)u | v(cid:11)Rn.
n
1