Expansion in finite simple groups of Lie type

The beautiful book of Terry Tao starts with the following words: Expander graphs are a remarkable type of graph (or more precisely, a family of graphs) on finite sets of vertices that manage to simultaneously be both sparse (low-degree) and “highly connected” at the same time. They enjoy very strong mixing properties: if one starts at a fixed vertex of an (two-sided) expander graph and randomly traverses its edges, then the distribution of one’s location will converge exponentially fast to the uniform distribution. For this and many other reasons, expander graphs are useful in a wide variety of areas of both pure and applied mathematics. Indeed, expander graphs have emerged as the area with the most fruitful interactions between computer science and pure mathematics. In computer science, expanders appear everywhere as basic building blocks of (communication) networks, in algorithms, derandomization, error correcting codes, and much more. The reader is referred to [HLW] for an excellent survey (though in the last decade since it was written, so much more has been done that an updated version will be a welcome addition to the literature). The current book gives the story from a different angle: the importance of expander graphs in pure mathematics and the use of pure mathematics to further advance the theory of expanders. In this sense this book follows [L1], but so much has been done in the last twenty-five years, and the theory went to some totally unexpected directions, that except for some similarity in the early chapters, the books are very different. Reviewing this book gives an opportunity to describe the fascinating development this area has made in the last decades. In spite of (or maybe because) I am personally involved in this process, going over this book was, for me, a wonderful journey in a beautiful interdisciplinary mathematics. Let me share some of this history. Expander graph is a family {Xi}i=1 of finite k-regular graphs (k-fixed) such that there exists a fixed ε > 0 with the following property: for every i and every subset Y of Xi, with |Y | ≤ 12 |Xi|, |∂Y | ≥ ε|Y |, when ∂Y is the set of edges of Xi going from Y to its complement. The nontrivial part is that the graphs are sparse (k fixed) and “very connected” (ε fixed). The first to define them and to prove their existence was Pinsker, though recently it was discovered that they had already appeared in the work of Kolmogorov and Barzdin. Chapter 1 of the book covers these aspects. In any case it was Margulis who gave the first explicit and constructible examples. For this he used Kazhdan’s property (T ) from representation theory of Lie groups and their discrete subgroups: If Γ is a group with property (T ) generated by a finite symmetric set S, then the family of Cayley graphs Cay(Γ/N ;S), when N runs over the finite index normal subgroups of Γ, forms a family of k-regular expander graphs, with k = |S|. Recall that the Cayley graph of a group G with respect to a set of