The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(pi (1)(M)) denote the space of (conjugacy classes of) d iscrete faithful representations of pi (1)(M) into PSL2(C). The components of the interior MP(pi (1)(M)) of AH(pi (1)(M)) (as a subset of the appropri ate representation variety) are enumerated by the space A(M) of marked home omorphism types of oriented, compact, irreducible 3-manifolds homotopy equi valent to M. In this paper, we give a topological enumeration of the compon ents of the closure of MP(pi (1)(M)) and hence a conjectural topological en umeration of the components of AH(pi (1)(M)). We do so by; characterizing e xactly which changes of marked homeomorphism type can occur in the algebrai c limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(pi (1)(M)) has infinitely many components.