The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a
compact 3-manifold with boundary M sits inside the PSL_2(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