Relatively Hyperbolic Groups with Semistable Peripheral Subgroups

Abstract

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

37 pages 8 figures. This updated version is more general than the first version which only considered one-ended groups. The thesis of A. Dasgupta facilitated the upgrade