Biregular expanders and the Ramanujan Conjecture

Expander graphs are well connected yet sparse graphs. The expanssion property of a graph is governed by the second largest eigenvalue of the adjacency matrix. I will consider quotients of the Bruhat-Tits building of GL(n), n=2,3, and view them as graphs. In this context the relationship between regular expander graphs and the Ramanujan Conjecture is well understood and has led to the definition and construction of asymptotically optimal regular expanders called Ramanujan graphs. In this talk I will show that biregular bipartite graphs obtained from the Bruhat-Tits building of a form of U(3) whose representations satisfy the Ramanujan conjecture are indeed Ramanujan bigraphs (this is joint work with Dan Ciubotaru). TIme permitting, I will also introduce a beautiful and simple combinatorial construction of constant degree expanders which are close to being Ramanujan. This construction, called the zig-zag product of graphs, is due to Reingold, Vadhan and Wigderson.