Relatively Hyperbolic Groups have Semistabile Fundamental Group at Infinity

TL;DR

This paper proves that relatively hyperbolic groups with boundary conditions have semistable fundamental groups at infinity, extending previous results and providing examples and improvements in the theory of group boundaries.

Contribution

It establishes semistability of fundamental groups at infinity for relatively hyperbolic groups under boundary conditions, generalizing prior results and refining existing theorems.

Findings

Semistability holds if boundary has no cut point.

Conditions can be relaxed under mild assumptions.

Improves a result by Dahmani and Groves.

Abstract

Suppose $G$ is a 1-ended finitely generated group that is hyperbolic relative to P a finite collection of 1-ended finitely generated subgroups. Our main theorem states that if the boundary $\partial (G, P)$ has no cut point, then $G$ has semistable fundamental group at $\infty$. Under mild conditions on $G$ and the members of P the 1-ended hypotheses and the no cut point condition can be eliminated to obtain the same semistability conclusion. We give an example that shows our main result is somewhat optimal. Finally, we improve a "double dagge" result of F. Dahmani and D. Groves.