Deformation spaces of Kleinian surface groups are not locally connected

Abstract

For any closed surface $S$ of genus $g\ge 2$, we show that the deformation space $AH\left(S\times I\right)$ of marked hyperbolic $3$–manifolds homotopy equivalent to $S$ 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.