Relatively Hyperbolic Groups with Semistable Peripheral Subgroups

TL;DR

This paper proves that relatively hyperbolic groups with finitely generated, semistable peripheral subgroups also have semistable fundamental groups at infinity, extending previous results to cases with cut points in the boundary.

Contribution

It generalizes the semistability at infinity property to a broader class of relatively hyperbolic groups with cut points in their boundary.

Findings

Semistability at infinity holds for relatively hyperbolic groups with semistable peripheral subgroups.

The result applies even when the boundary contains cut points.

Extends known results to more general boundary topologies.

Abstract

Suppose $G$ is a finitely presented group that is hyperbolic relative to ${\bf P}$ a finite collection of 1-ended finitely generated proper subgroups of $G$. If $G$ and the ${\bf P}$ are 1-ended and the boundary $\partial (G,{\bf P})$ has no cut point, then $G$ was known to have semistable fundamental group at $\infty$. We consider the more general situation when $\partial (G,{\bf P})$ contains cut points. Our main theorem states that if $G$ is finitely presented and each $P\in {\bf P}$ is finitely generated and has semistable fundamental group at $\infty$, then $G$ has semistable fundamental group at $\infty$.