Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL2(C)-character variety X(M) of pi(1)(M). We study the dynamics of the action of Out (pi(1)(M)) on both AH(M) and X(M). The nature of the dynamics reflects the topology of M. The quotient AI(M) = AH(M)/Out(pi(1)(M)) may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to M and its topology reflects the dynamics of the action.