Deformation spaces of Kleinian surface groups are not locally connected

TL;DR

This paper proves that the deformation space of hyperbolic 3-manifolds associated with closed surfaces of genus at least 2 is not locally connected, extending Bromberg's earlier results on punctured torus groups.

Contribution

It establishes the non-local connectivity of the deformation space for closed surface groups, using a new filling theorem based on cone-manifold deformation theory.

Findings

Deformation space $AH(S imes I)$ is not locally connected.

Extension of Bromberg's non-local connectivity result to closed surfaces.

Introduction of a new filling theorem based on cone-manifold deformations.

Abstract

For any closed surface $S$ of genus $g \geq 2$, we show that the deformation space of marked hyperbolic 3-manifolds homotopy equivalent to $S$, $AH(S \times I)$, is not locally connected. This proves a conjecture of Bromberg who recently proved that the space of Kleinian punctured torus groups is not locally connected. Playing an essential role in our proof is a new version of the filling theorem that is based on the theory of cone-manifold deformations developed by Hodgson, Kerckhoff, and Bromberg.