Kazhdan's Property (T) for Graphs

Abstract

D. A. Kahzdan first put forth property (T) in relation to the study of discrete subgroups of Lie groups of finite co-volume. Through a combinatorial approach, we define an analogue of property (T) for regular graphs. We then prove the basic combinatorial and metric properties of Kazhdan groups in this context. In particular, we use our methods to construct infinite families of expanders as in the classical case. Finally, we consider the combinatorial analogue of the group theoretic property $(τ)$ and prove its basic properties.