Expansion in Finite Simple Groups of Lie Type: (Graduate Studies in Mathematics)

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-Szemeredi-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.

Table of Contents:
Expansion in Cayley graphs Expander graphs: Basic theory Expansion in Cayley graphs, and Kazhdan's property (T) Quasirandom groups The Balog-Szemeredi-Gowers lemma, and the Bourgain-Gamburd expansion machine Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality Non-concentration in subgroups Sieving and expanders Related articles Cayley graphs the algebra of groups The Lang-Weil bound The spectral theorem and its converses for unbounded self-adjoint operators Notes on Lie algebras Notes on groups of Lie type Bibliography Index

About the Author :
Terence Tao was the winner of the 2014 Breakthrough Prize in Mathematics. He is the James and Carol Collins Chair of mathematics at UCLA and the youngest person ever to be promoted to full professor at the age of 24. In 2006 Tao became the youngest ever mathematician to win the Fields Medal. His other honours include the George Polya Prize from the Society for Industrial and Applied Mathematics (2010), the Alan T Waterman Award from the National Science Foundation (2008), the SASTRA Ramanujan Prize (2006), the Clay Research Award from the Clay Mathematical Institute (2003), the Bocher Memorial Prize from the American Mathematical Society (2002) and the Salem Prize (2000).

Review :
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, Jahresbericht der Deutschen Mathematiker-Vereinigung