Uniform Kazhdan Constant for {SLn(Z)}n≥3

Let R be a ring generated by l elements with stable range r. Assume that the group EL d (R) has Kazhdan constant ǫ 0 > 0 for some d ≥ r + 1. We prove that there exist ǫ(ǫ 0 , l) > 0 and k ∈ N, s.t. for every n ≥ d, EL n (R) has a generating set of order k and a Kazhdan constant larger than ǫ. As a consequence, we obtain for SL n (Z) where n ≥ 3, a Kazhdan constant which is independent of n w.r.t generating set of a fixed size.