Dynamics on PSL(2,C)-character varieties of certain hyperbolic 3-manifolds

Let M be a compact orientable hyperbolizable 3-manifold. In this thesis we study the action of Out(pi1(M)) the group of outer automorphisms of the fundamental group of M on X (pi1(M), PSL(2, C )) the PSL(2, C )-character variety of pi1(M). In particular, we find a domain of discontinuity U for this action when M is a hyperbolizable twisted I-bundle and when M is a hyperbolizable nontrivial compression body without toroidal boundary components. Our domain of discontinuity is strictly larger than the previously known domain of discontinuity and contains both discrete and faithful and dense representations. Moreover, in the case when M is a twisted I-bundle we show that U is maximal with respect to the discrete and faithful representations.