Local connectedness of boundaries for relatively hyperbolic groups

Abstract

Let $(\hat{I}“,\mathbb{P})$ be a relatively hyperbolic group pair that is relatively one ended. Then the Bowditch boundary of $(\hat{I}“,\mathbb{P})$ 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 $\hat{I}“$ and no restriction on the peripheral subgroups $P\in \mathbb{P}$.

39 pages. Version 2 includes several clarifications in response to feedback from the referee. To appear in the Journal of Topology