**Graduate Studies in Mathematics**

Volume: 164;
2015;
303 pp;
Hardcover

MSC: Primary 05; 11; 20;

**Print ISBN: 978-1-4704-2196-0
Product Code: GSM/164**

List Price: $84.00

AMS Member Price: $67.20

MAA Member Price: $75.60

**Electronic ISBN: 978-1-4704-2265-3
Product Code: GSM/164.E**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

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#### Supplemental Materials

# Expansion in Finite Simple Groups of Lie Type

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*Terence Tao*

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 expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. 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 additional 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.

#### Readership

Graduate students and research mathematicians interested in graph theory, geometric group theory, and arithmetic combinatorics.

#### Reviews & Endorsements

Asymptotic group theory is a recently emerging branch of group theory, that can be described as the study of groups whose order is finite --- but large! Tao's book is certainly a valuable introduction to that exciting new subject.

-- Alain Valette, Jahresber Dtsch Math-Ver

#### Table of Contents

# Table of Contents

## Expansion in Finite Simple Groups of Lie Type

- Cover Cover11
- Title iii4
- Dedication v6
- Contents vii8
- Preface xi12
- Part 1. Expansion in Cayley graphs 116
- Chapter 1. Expander graphs: Basic theory 318
- Chapter 2. Expansion in Cayley graphs, and Kazhdan’s property (T) 2338
- Chapter 3. Quasirandom groups 5772
- Chapter 4. The Balog-Szemerédi-Gowers lemma, and the Bourgain-Gamburd expansion machine 85100
- Chapter 5. Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality 101116
- Chapter 6. Non-concentration in subgroups 135150
- Chapter 7. Sieving and expanders 143158
- Part 2. Related articles 165180
- Chapter 8. Cayley graphs the algebra of groups 167182
- Chapter 9. The Lang-Weil bound 187202
- Chapter 10. The spectral theorem and its converses for unbounded self-adjoint operators 203218
- Chapter 11. Notes on Lie algebras 227242
- Chapter 12. Notes on groups of Lie type 267282
- Bibliography 293308
- Index 301316
- Back Cover Back Cover1319