Abstract: The Cheeger constant of a graph is a measure of the graph's resiliency. In a graph with a small Cheeger constant the deletion of just a few edges is sufficient to split the graph into two pieces of close to the same size. Since its introduction in the late 1970s, it has become an essential tool in the study of the graph's eigenvalue spectrum. An especially intriguing application arises from a connection between graphs and Riemann surfaces. In some cases, geometric questions about the surfaces can be answered by studying the connectivity properties of associated graphs. In the first half of this talk we shall present some basic material about the Cheeger constant and its uses in graph theory. Then, in the second half, we shall consider an application of these ideas to Cayley graphs of certain matrix groups. We will show how studying these particular graphs is relevant to questions arising from arithmetic Riemann surfaces. This area of research provides a nice synthesis of ideas from combinatorics, geometry and number theory. |