Random walks and spectral gaps in linear groups

Let S ⊂ SLd(Z ) be a finite and symmetric set, i.e. we assume that whenever gamma ∈ S, gamma-1 ∈ S, too. In this thesis we would like to understand the distribution of words of a given length l formed from the elements of S in congruence classes modulo an integer q. More specifically we are interested in how large l is needed to be taken for a given q, so that the words are "almost uniformly distributed" in the congruence classes. We will state and prove results which implies this for l << log q.;These results have applications in sieving in number theory and group theory, and they are also related to certain properties of covering spaces of hyperbolic 3-manifolds.