American Mathematical Society

Terence Tao

Graduate Studies in Mathematics

Volume 164

Expansion in Finite Simple Groups of Lie Type

Expansion in Finite Simple Groups of Lie Type

Expansion in Finite Simple Groups of Lie Type

Terence Tao

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 164

https://doi.org/10.1090//gsm/164

EDITORIAL COMMITTEE

Dan AbramovichDaniel S. Freed

Rafe Mazzeo (Chair)Gigliola Staffilani

2010 Mathematics Subject Classification. Primary 05C81, 11B30, 20C33, 20D06, 20G40.

For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-164

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Tao, Terence, 1975Expansion in finite simple groups of Lie type / Terence Tao.

pages cm. – (Graduate studies in mathematics ; volume 164)Includes bibliographical references and index.ISBN 978-1-4704-2196-0 (alk. paper)1. Finite simple groups. 2. Lie groups. I. Title.

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Contents

Preface xi

Notation xii

Acknowledgments xiii

Part 1. Expansion in Cayley Graphs

Chapter 1. Expander graphs: Basic theory 3

§1.1. Expander graphs 4

§1.2. Connection with edge expansion 9

§1.3. Random walks on expanders 15

§1.4. Random graphs as expanders 17

Chapter 2. Expansion in Cayley graphs, and Kazhdan’s property (T) 23

§2.1. Kazhdan’s property (T) 27

§2.2. Induced representations and property (T) 37

§2.3. The special linear group and property (T) 47

§2.4. A more elementary approach 55

Chapter 3. Quasirandom groups 57

§3.1. Mixing in quasirandom groups 62

§3.2. An algebraic description of quasirandomness 67

§3.3. A weak form of Selberg’s 3/16 theorem 67

Chapter 4. The Balog-Szemeredi-Gowers lemma, and the Bourgain-Gamburd expansion machine 85

§4.1. The Balog-Szemeredi-Gowers lemma 87

vii

viii Contents

§4.2. The Bourgain-Gamburd expansion machine 97

Chapter 5. Product theorems, pivot arguments, and the Larsen-Pinknonconcentration inequality 101

§5.1. The sum-product theorem 104

§5.2. Finite subgroups of SL2 110

§5.3. The product theorem in SL2(k) 120

§5.4. The product theorem in SLd(k) 125

§5.5. Proof of the Larsen-Pink inequality 129

Chapter 6. Nonconcentration in subgroups 135

§6.1. Expansion in thin subgroups 137

§6.2. Random generators expand 140

Chapter 7. Sieving and expanders 143

§7.1. Combinatorial sieving 146

§7.2. The strong approximation property 156

§7.3. Sieving in thin groups 160

Part 2. Related Articles

Chapter 8. Cayley graphs and the algebra of groups 167

§8.1. A Hall-Witt identity for 2-cocycles 177

Chapter 9. The Lang-Weil bound 187

§9.1. The Stepanov-Bombieri proof of the Hasse-Weil bound 194

§9.2. The proof of the Lang-Weil bound 198

§9.3. Lang-Weil with parameters 200

Chapter 10. The spectral theorem and its converses for unboundedself-adjoint operators 203

§10.1. Self-adjointness and resolvents 207

§10.2. Self-adjointness and spectral measure 212

§10.3. Self-adjointness and flows 218

§10.4. Essential self-adjointness of the Laplace-Beltrami operator 224

Chapter 11. Notes on Lie algebras 227

§11.1. Abelian representations 233

§11.2. Engel’s theorem and Lie’s theorem 235

§11.3. Characterising semisimplicity 237

§11.4. Cartan subalgebras 242

Contents ix

§11.5. sl2 representations 245

§11.6. Root spaces 247

§11.7. Classification of root systems 251

§11.8. Chevalley bases 258

§11.9. Casimirs and complete reducibility 263

Chapter 12. Notes on groups of Lie type 267

§12.1. Simple Lie groups over C 268

§12.2. Chevalley groups 278

§12.3. Finite simple groups of Lie type 288

Bibliography 293

Index 301

Preface

Expander graphs are a remarkable type of graph (or more precisely, a familyof graphs) on finite sets of vertices that manage to simultaneously be bothsparse (low-degree) and “highly connected” at the same time. They enjoyvery strong mixing properties: if one starts at a fixed vertex of an (two-sided)expander graph and randomly traverses its edges, then the distribution ofone’s location will converge exponentially fast to the uniform distribution.For this and many other reasons, expander graphs are useful in a wide varietyof areas of both pure and applied mathematics.

There are now many ways to construct expander graphs, but one of theearliest constructions was based on the Cayley graphs of a finite group (or ofa finitely generated group acting on a finite set). The expansion property forsuch graphs turns out to be related to a rich variety of topics in group the-ory and representation theory, including Kazhdan’s property (T), Gowers’notion of a quasirandom group, the sum-product phenomenon in arithmeticcombinatorics, and the Larsen-Pink classification of finite subgroups of alinear group. Expansion properties of Cayley graphs have also been appliedin analytic number theory through what is now known as the affine sieve ofBourgain, Gamburd, and Sarnak, which can count almost prime points inthin groups.

This text is based on the lecture notes from a graduate course on thesetopics I gave at UCLA in the winter of 2012, as well as from some additionalposts on my blog at terrytao.wordpress.com on further related topics.The first part of this text can thus serve as the basis for a one-quarteror one-semester advanced graduate course, depending on how much of theoptional material one wishes to cover. While the material here is largelyself-contained, some basic graduate real analysis (in particular, measure

xi

xii Preface

theory, Hilbert space theory, and the theory of Lp norms), graph theory,and linear algebra (e.g., the spectral theorem for unitary matrices) will beassumed. Some prior familiarity with the classical Lie groups (particularlythe special linear group SLn and the unitary group Un) and representationtheory will be helpful but not absolutely necessary. To follow Section 3.3(which is optional) some prior exposure to Riemannian geometry would alsobe useful.

The core of the text is Part 1. After discussing the general theory ofexpander graphs in the first section, we then specialise to the case of Cayleygraphs, starting with the remarkable observation1 of Margulis linking Kazh-dan’s property (T) with expansion, and then turning to the more recentobservations of Sarnak, Xue, Gamburd, and Bourgain linking the propertyof finite groups now known as quasirandomness with expansion, which isalso related to the famous “3/16 theorem” of Selberg. As we will presentin this text, this sets up a general “machine” introduced by Bourgain andGamburd for verifying expansion in a Cayley graph, which in addition toquasirandomness requires two additional ingredients, namely a product the-orem and a nonconcentration estimate. These two ingredients are then thefocus of the next two sections of this part. The former ingredient uses tech-niques from arithmetic combinatorics related to the sum-product theorem,as well as estimates of Larsen and Pink on controlling the interaction be-tween finite subgroups of a linear group and various algebraic varieties (suchas conjugacy classes or maximal tori). The latter ingredient is perhaps themost delicate aspect of the theory, and often requires a detailed knowledgeof the algebraic (and geometric) structure of the ambient group. Finally, wepresent an application of these ideas to number theory by introducing thebasics of sieve theory, and showing how expansion results may be insertedinto standard sieves to give new bounds on almost primes in thin groups.

Part 2 contains a variety of additional material that is related to oneor more of the topics covered in Part 1, but which can be omitted for thepurposes of teaching a graduate course on the subject.

Notation

For reasons of space, we will not be able to define every single mathematicalterm that we use in this book. If a term is italicised for reasons other thanemphasis or for definition, then it denotes a standard mathematical object,result, or concept, which can be easily looked up in any number of references.

1This material in Section 2 is not absolutely required for subsequent sections of this part,although it does provide some helpful context for these later sections. Thus, this section may beabridged or even omitted altogether in a lecture course if desired.

Acknowledgments xiii

(In the blog version of the book, many of these terms were linked to theirWikipedia pages, or other on-line reference pages.)

Given a subset E of a space X, the indicator function 1E : X → R isdefined by setting 1E(x) equal to 1 for x ∈ E and equal to 0 for x �∈ E.

The cardinality of a finite set E will be denoted |E|. We will use2 theasymptotic notation X = O(Y ), X � Y , or Y � X denote the estimate|X| ≤ CY for some absolute constant C > 0. In some cases we will needthis constant C to depend on a parameter (e.g., d), in which case we shallindicate this dependence by subscripts, e.g., X = Od(Y ) or X �d Y . Wealso sometimes use X ∼ Y as a synonym for X � Y � X. If n is aparameter going to infinity, we let on→∞(1) denote a quantity depending onn and bounded in magnitude by c(n) for some quantity c(n) that goes tozero as n → ∞. More generally, given an additional parameter such as k, welet on→∞;k(1) denote a quantity that may depend on both k and n, whichis bounded by ck(n) for some quantity ck(n) that goes to zero as n → ∞ foreach fixed k.

Acknowledgments

I am greatly indebted to my students of the course on which this textwas based, as well as many further commenters on my blog, including IanAgol, Abhishek Bhowmick, Nick Cook, Sean Eberhard, Alireza Golsefidy,Joerg Grande, Ben Green, Dick Gross, Harald Helfgott, Arie Israel, DavidJoyner, Matthew Kahle, Emmanuel Kowalski, Wolfgang Moens, Vipul Niak,William Orrick, Mikhail Ostrovskii, David Roberts, Misha Rudnev, Alexan-der Shaposhnikov, Lior Silberman, Vit Tucek, Yilong Yang, and Wei Zhou.These comments can be viewed online at:

terrytao.wordpress.com/category/teaching/254b-expansion-in-groups/

The author was supported by the NSF grant DMS-0649473, the Jamesand Carol Collins Chair, the Mathematical Analysis & Application ResearchFund Endowment, and the Simons Foundation.

2Once we deploy the machinery of nonstandard analysis in Section 6, we will use a closelyrelated, but slightly different, asymptotic notation.

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Index

(B,N)-pair, 283

abelianisation, 230adjacency matrix, 5adjoint, 208adjoint form, 271adjoint representation, 231affine algebraic variety, 187almost prime, 144approximate group, 86asymptotic notation, xiiiautocorrelation function, 50

Balog-Szemeredi theorem, 91Balog-Szemeredi-Gowers lemma

approximate group form, 93product set form, 92

Bertini’s theorem, 198beta sieve, 154bipartite graph, 88Bochner’s theorem, 49Bonferroni inequalities, 19Borel subgroup, 111, 281bounded functional calculus, 214Bourgain-Gamburd expansion machine,

87Bruhat decomposition, 283Brun’s theorem, 155

Cartan decomposition, 247Cartan matrix, 261Cartan semisimplicity criterion, 232Cartan solvability criterion, 240Cartan subalgebra, 242

Cartan’s theorem, 270

Casimir operator, 264

Cayley graph, 23, 167

Cayley transform, 69

centre of a Lie algebra, 231

characteristic subalgebra, 239

Cheeger constant, 10

Chevalley basis, 263

Chevalley group, 281

Chevalley normalisation, 262

chromatic number, 14

class equation, 117

classical Lie algebra, 260

closure of an operator, 207

coboundary, 183

cochain, 177

cocycle, 35, 179

combinatorial sieve, 150

compact form, 272

complemented subspace, 241

complete graph, 4, 8

complexity of an algebraic variety, 187

composition factors, 157

composition series, 157

concentration of measure, 15

concrete Lie algebra, 228

conjugacy class, 114

conjugacy of Cartans, 244

Connectivity of an expander graph, 14

converse to Balog-Szemeredi-Gowers, 95

convolution, 62

coroot, 249

coweight lattice, 276

301

302 Index

Coxeter diagram, 253cusp, 70cyclic vector, 217

depolarisation, 214derivation, 229derived algebra, 230derived series, 230diagram, 169Diameter of an expander graph, 14dimension, 126dimension of an algebraic variety, 187direct product, 229direct sum, 229direct sum of representations, 29discrete Cheeger inequality, 13discrete Cheeger inequality, weak, 10Dynkin diagram, 255

edge expansion, 9Engel’s theorem, 236escape from subspaces, 121essentially self-adjoint operator, 209exceptional Lie algebra, 260expander family, 6expander mixing lemma, 13extended Dynkin diagram, 257

flattening lemma, 83, 98Fokker-Planck equation, 81Frobenius endomorphism, 190Frobenius lemma, 59functional calculus, 203fundamental domain, 70

generalised eigenspace, 233girth, 14Goursat’s lemma, 159graph, 4graph Laplacian, 6graph metric, 14

Hardy’s inequality, 73Hasse-Weil bound, 190heat flow, 218Hellinger-Toeplitz theorem, 209Herglotz representation theorem, 213horocycle flow, 54hyperbolic plane, 68

indecomposable representation, 241independent set, 14indicator function, xiii

induced representation, 39invariant subspace, 217invariant vector, 30involved torus, 124irreducible representaiton, 57irreducible representation, 241irreducible root system, 250isogeny, 286

Jacobi identity, 227Jacobi operator, 212Jordan’s theorem, 67Jordan-Chevalley decomposition, 234Jordan-Holder theorem, 157

Katz-Tao lemma, 107Kazhdan constant, 30Killing form, 231

Landau conjectures, 143, 155Lang-Weil bound, 189Lang-Weil with parameters, 192large sieve, 156Larsen-Pink inequality, 113, 115, 122Lefschetz principle, 140Legendre identity, 148Legendre sieve, 148Leibniz rule, 229Levi decomposition, 232Lie algebra, 227Lie algebra ideal, 229Lie’s second theorem, 269Lie’s theorem, 237Lie’s third theorem, 269linear, 45Littlewood-Paley projection, 223lower central series, 230Lubotzky’s 1-2-3 problem, 136

Mobius function, 148Mautner phenomenon, 53maximal torus, 111, 281Mersenne prime, 143mixing inequality, 63modular curve, 193Moore ergodic theorem, 54multiplicative energy, 92

nilpotent Lie algebra, 230nilpotent operator, 234nonconcentration estimate, 87normaliser, 242notation, xii

Index 303

one-sided expander, 6

parabolic subgroup, 284perfect group, 59perfect Lie algebra, 230pivot argument, 106Poincare disk, 69Poincare half-plane, 68Poincare inequality, 9polar decomposition, 272polarisation identity, 51polycyclic Lie algebra, 231primorial, 148principal congruence subgroup, 74principal modular curve, 70principal series representation, 60product theorem, 87, 101, 120

quasirandom group, 57quasiregular representation, 29quasisimple group, 271

radical of a Lie algebra, 232Ramanujan graph, 7rank of a Lie algebra, 244Rayleigh quotient, 72Ree groups, 290regular element, 277regular element of a Lie algebra, 242regular graph, 4regular representation, 29regular semisimple element, 111regular unipotent element, 111representation (of a Lie algebra), 228resolvent, 210Resolvent identity, 210root system, 250root vector, 248Ruzsa covering lemma, 95, 107Ruzsa triangle inequality, 93, 107

Schrodinger propagator, 220Schreier graph, 24Schwarz-Zippel type bound, 188Selberg sieve, 156Selberg’s 3/16 theorem, 75Selberg’s conjecture, 74Selberg’s expander construction, 68self-adjoint operator, 209semidirect product, 229semisimple Jordan decomposition, 241semisimple Lie algebra, 232semisimple operator, 234

short exact sequence, 228simple Lie algebra, 231simple Lie group, 271solvable Lie algebra, 230spectral gap, 71split extension, 229Steinberg group, 290Steinberg representation, 60Stone’s theorem, 220strong approximation property, 159subrepresentation, 30sum-product theorem, 105sumset estimates, 107Suzuki groups, 290

Tits alternative, 138transitivity of induction, 40trivial representation, 28twin prime, 143twisted group of Lie type, 291two-sided expander, 6

unipotent group, 111unitary representation, 28universal cover, 269

variety, 126virtually quasirandom group, 66

wedding cake decomposition, 12weight vector, 235weighted Balog-Szemeredi-Gowers

lemma, 86weighted Balog-Szemeredi-Gowers

theorem, 91Weyl alcove, 277Weyl chamber, 258Weyl group, 257Weyl’s complete reducibility theorem,

232, 263Whitehead’s lemma, 264

GSM/164

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Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expan-sion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combina-torics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan’s property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog–Szemerédi–Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with addi-tional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang–Weil bound, as well as numerous exercises and other optional material.

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