Examples of relative deformation spaces that are not locally connected

TL;DR

This paper constructs an infinite family of pared manifolds demonstrating that their relative deformation spaces of hyperbolic structures can lack local connectivity, extending previous findings about Kleinian punctured torus groups.

Contribution

It introduces new examples of pared manifolds with non-locally connected deformation spaces, broadening understanding of hyperbolic structure moduli spaces.

Findings

Existence of infinite families of pared manifolds with non-locally connected deformation spaces

Extension of Bromberg's result on punctured torus groups

Demonstration of complex topology in hyperbolic structure spaces

Abstract

We provide an infinite family of pared manifolds whose relative deformation spaces of hyperbolic structures on these manifolds are not locally connected. This is a natural extension of the recent result of Bromberg that shows the space of Kleinian punctured torus groups is not locally connected.