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Hyperbolic groups and local connectivity

G. Christopher Hruska, Kim Ruane

TL;DR

The paper analyzes the boundary geometry of one-ended hyperbolic groups, explaining Bestvina–Mess's local connectivity result and delivering elementary proofs of semistability at infinity and linear connectivity via the $$(\ddag_M)$$ property. It shows that failure of $$(\ddag_M)$$ would force a separating horoball and a cut point, which Bowditch's Cut Point Theorem excludes, thereby establishing the property for all $M>0$ in the one-ended setting and deriving semistability. It then derives linear connectivity of the boundary through a constructive, elementary argument and situates these results within the broader framework, including extensions to relatively hyperbolic settings and the Bonk–Kleiner quasi-hyperbolic plane theorem. Collectively, the work provides accessible proofs that strengthen the understanding of boundary geometry and its implications for the global structure of hyperbolic groups.

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.

Hyperbolic groups and local connectivity

TL;DR

The paper analyzes the boundary geometry of one-ended hyperbolic groups, explaining Bestvina–Mess's local connectivity result and delivering elementary proofs of semistability at infinity and linear connectivity via the property. It shows that failure of would force a separating horoball and a cut point, which Bowditch's Cut Point Theorem excludes, thereby establishing the property for all in the one-ended setting and deriving semistability. It then derives linear connectivity of the boundary through a constructive, elementary argument and situates these results within the broader framework, including extensions to relatively hyperbolic settings and the Bonk–Kleiner quasi-hyperbolic plane theorem. Collectively, the work provides accessible proofs that strengthen the understanding of boundary geometry and its implications for the global structure of hyperbolic groups.

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.
Paper Structure (6 sections, 16 theorems, 23 equations)

This paper contains 6 sections, 16 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. $\delta$--hyperbolic spaces
  4. Separating horoballs and global cut points
  5. Semistability at infinity
  6. Linear connectivity

Key Result

Proposition 2.2

If $X$ is $\delta$--hyperbolic, then for any base point $p \in X$ and any $x,y,z\in X$.

Theorems & Definitions (42)

  • Definition 2.1: Hyperbolic spaces
  • Proposition 2.2: BuyaloSchroeder07, Prop. 2.1.2
  • Definition 2.3: The set $\overline{X}=X \cup \partial X$
  • Definition 2.4: The topology on $\overline{X}$
  • Definition 2.5: Radial paths
  • Definition 2.6: Extended Gromov product
  • Lemma 2.7
  • proof
  • Remark 2.8: Slim ideal triangles
  • Definition 2.9: Visual metric
  • ...and 32 more