Walks on graphs and lattices -- effective bounds and applications

We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Gamma. We consider all walks of length N on G, starting from v_i and ending at v_j To each such walk $w$ we assign the element of Gamma equal to the product of the elements along the walk. The set of all walks of length N from v_i to v_j thus induces a probability distribution $F_N on Gamma In previous work we have given necessary and sufficient conditions for the limit as N goes to infinity of F_N to exist and to be the uniform density on Gamma. The convergence speed is then exponential in N. In this paper we consider (G, Gamma) where Gamma is a group possessing Kazhdan's property T (or, less restrictively, property tau with respect to representations with finite image), and a family of homomorphisms\psi_k: Gamma -> Gamma_k with finite image. Each F_N induces a distribution$F_{N, k} on Gamma_k (by push-forward). Our main result is that, under mild technical assumptions, the exponential rate of convergence of $F_{N, k} to the uniform distribution on Gamma_k does not depend on k. As an application, we prove effective versions of the results of the author on the probability that a random (in a suitable sence) element of SL(n, Z) or Sp(n, Z) has irreducible characteristic polynomial, generic Galois group, etc.