Title: Expansion in finite simple groups of Lie type
Description: 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.
Subjects: Finite simple groups, Lie groups, Combinatorics -- Graph theory -- Random walks on graphs, Number theory -- Sequences and sets -- Arithmetic combinatorics; higher degree uniformity, Group theory and generalizations -- Representation theory of groups -- Representations of finite groups of Lie type, Group theory and generalizations -- Abstract finite groups -- Simple groups: alternating groups and groups of Lie type, Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups over finite fields