Random Walks in Linear Groups

I will talk about a joint work with Jean Bourgain that establishes spectral gaps for random walks on SL_n(Z/qZ). Let S be a fixed finite and symmetric subset of SL_n(Z) which generates a Zariski dense subgroup. We show that words of length C log(q) are almost uniformly distributed among congruence classes modulo q. Unlike in previous results, q is arbitrary and not restricted to any special subset of the integers. A key new ingredient for the proof is the recent work of Bourgain, Furman, Lindenstrauss and Mozes on the equidistribution of non-Abelian semigroup actions on the torus.