Abstract
Let S subset of GL(n)(Z) be a finite symmetric set. We show that if the Zariski closure of Gamma < S > is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay (Gamma/Gamma(q), pi(q)(S)) is O(log q), where q is an arbitrary positive integer, pi(q) : Gamma -> Gamma/Gamma(q) is the canonical projection induced by the reduction modulo q, and the implied constant depends only on S.