Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group ELd(R) has Kazhdan constant ϵ0>0 for some d⩾r+1. We prove that there exist ϵ(ϵ0,l)>0 and k∈N, s.t. for every n⩾d, ELn(R) has a generating set of order k and a Kazhdan constant larger than ϵ. As a consequence, we obtain for SLn(Z) where n⩾3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.