Relatively Hyperbolic Groups have Semistabile Fundamental Group at Infinity

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.