Hardcover ISBN:  9781470421960 
Product Code:  GSM/164 
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Electronic ISBN:  9781470422653 
Product Code:  GSM/164.E 
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Book DetailsGraduate Studies in MathematicsVolume: 164; 2015; 303 ppMSC: Primary 05; 11; 20;
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 selfcontained, 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.
ReadershipGraduate students and research mathematicians interested in graph theory, geometric group theory, and arithmetic combinatorics.

Table of Contents

Part 1. Expansion in Cayley graphs

Chapter 1. Expander graphs: Basic theory

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

Chapter 3. Quasirandom groups

Chapter 4. The BalogSzemerédiGowers lemma, and the BourgainGamburd expansion machine

Chapter 5. Product theorems, pivot arguments, and the LarsenPink nonconcentration inequality

Chapter 6. Nonconcentration in subgroups

Chapter 7. Sieving and expanders

Part 2. Related articles

Chapter 8. Cayley graphs the algebra of groups

Chapter 9. The LangWeil bound

Chapter 10. The spectral theorem and its converses for unbounded selfadjoint operators

Chapter 11. Notes on Lie algebras

Chapter 12. Notes on groups of Lie type


Additional Material

Reviews

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 MathVer


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

Part 1. Expansion in Cayley graphs

Chapter 1. Expander graphs: Basic theory

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

Chapter 3. Quasirandom groups

Chapter 4. The BalogSzemerédiGowers lemma, and the BourgainGamburd expansion machine

Chapter 5. Product theorems, pivot arguments, and the LarsenPink nonconcentration inequality

Chapter 6. Nonconcentration in subgroups

Chapter 7. Sieving and expanders

Part 2. Related articles

Chapter 8. Cayley graphs the algebra of groups

Chapter 9. The LangWeil bound

Chapter 10. The spectral theorem and its converses for unbounded selfadjoint operators

Chapter 11. Notes on Lie algebras

Chapter 12. Notes on groups of Lie type

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 MathVer