The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π1(M)) denote the space of (conjugacy classes of) discrete faithful representations of π1(M) into PSL2(C). The components of the interior MP(π1(M)) of AH(π1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M . In this paper, we give a topological enumeration of the components of the closure of MP(π1(M)) and hence a conjectural topological enumeration of the components of AH(π1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifoldsM for which AH(π1(M)) has infinitely many components.