Local topology in deformation spaces of hyperbolic 3-manifolds

TL;DR

This paper proves local connectivity of deformation spaces of hyperbolic 3-manifolds at specific points, enhancing understanding of their topological structure and behavior near special types of points.

Contribution

It establishes local connectivity of deformation spaces at minimally parabolic and quasiconformally rigid points, extending previous results to new classes of 3-manifolds.

Findings

Deformation space AH(M) is locally connected at minimally parabolic points.

Spaces of Kleinian surface groups are locally connected at quasiconformally rigid points.

Similar local connectivity results hold for acylindrical 3-manifolds and Bers slices.

Abstract

We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.