Local connectedness of boundaries for relatively hyperbolic groups

TL;DR

This paper proves that the Bowditch boundary of relatively hyperbolic groups with a single end is locally connected, removing previous restrictions on peripheral subgroup properties and group cardinality.

Contribution

It generalizes prior results by establishing local connectedness without restrictions on peripheral subgroup structure or group size.

Findings

Bowditch boundary is locally connected for relatively one-ended hyperbolic groups

No restrictions on peripheral subgroup finiteness or torsion

Results apply to groups of arbitrary cardinality

Abstract

Let $(\Gamma,\mathbb{P})$ be a relatively hyperbolic group pair that is relatively one ended. Then the Bowditch boundary of $(\Gamma,\mathbb{P})$ is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of $\Gamma$ and no restriction on the peripheral subgroups $P \in \mathbb{P}$.