The topology of deformation spaces of Kleinian groups

Abstract

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π_1(M)) denote the space of (conjugacy classes of) discrete faithful representations of π_1(M) into PSL 2 (C). The components of the interior MP(π_1(M)) of AH(π_1(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(π_1(M)) and hence a conjectural topological enumeration of the components of AH(π_1(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 manifolds M for which AH(π_1(M)) has infinitely many components.