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) discrete faithful representations of $\pi_1(M)$ into $\mathrm{PSL}_2(\mathbf{C})$. The components of the interior $MP(\pi_1(M))$ of $AH(\pi_1(M))$ (as a subset of the appropriate representation variety) are enumerated by the space $\mathcal{A}(M)$ 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 $MP(\pi_1(M)$ and hence a conjectural topological enumeration of the components of $AH(\pi_1(M))$. We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a 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.



James W. Anderson

Richard D. Canary

Darryl McCullough