Kazhdan Constants of Hyperbolic Groups

Let H be an infinite hyperbolic group with Kazhdan property (T ) and let κ(H,X) denote 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.