  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.
