Hyperbolic groups and local connectivity

TL;DR

This paper explains key results on the local connectivity and boundary properties of one-ended hyperbolic groups, providing elementary proofs and connecting shape theory with geometric analysis.

Contribution

It offers elementary proofs of local connectivity and linear connectivity of hyperbolic group boundaries, building on Bestvina-Mess's results and unifying different approaches.

Findings

Hyperbolic groups have locally connected boundaries.

All hyperbolic groups are semistable at infinity.

Boundaries of one-ended hyperbolic groups are linearly connected.

Abstract

The goal of this paper is to give an exposition of some results of Bestvina-Mess on local connectivity of the boundary of a one-ended word hyperbolic group. We also give elementary proofs that all hyperbolic groups are semistable at infinity and their boundaries are linearly connected in the one-ended case. Geoghegan first observed that semistability at infinity is a consequence of local connectivity using ideas from shape theory, and Bonk-Kleiner proved linear connectivity using analytical methods. The methods in this paper are closely based on the original ideas of Bestvina-Mess.