Kazhdan Constants of Hyperbolic Groups

Abstract

Let H be an infinite hyperbolic group with Kazhdan property (T ) and let κ(H, X) de- note the Kazhdan constant of H with respect to a generating set X. We prove that infX κ(H, X )=0 , where the infimum is taken over all finite generating sets of H. In particular,this gives an answer to a Lubotzky question.