Lie Groups, Representations and Discrete Mathematics

For a discrete group G and a finite subset X of G, let K(G, X) denote the Kazhdan constant of G associated to X. We define the uniform Kazhdan constant of G by K(G) = min { K(G,X) | X is finite and generates G }. Obviously K(G)>0 for any finite group G. On the other hand, K(G)=0 for many infinite Kazhdan groups G and, moreover, the question of the existence of an infinite group with non--zero uniform Kazhdan constant was open until now. The main goal of this talk is to provide the affirmative answer. This is a joint work with Dmitry Sonkin.