Local cut points and splittings of relatively hyperbolic groups

Abstract

We show that the existence of a nonparabolic local cut point in the Bowditch boundary $\partial \left(G,\mathbb{P}\right)$ of a relatively hyperbolic group $\left(G,\mathbb{P}\right)$ implies that $G$ splits over a $2$–ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of $1$–dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over $2$–ended subgroups and no peripheral splittings.

In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua $X$ and $Y$, then $Ends\left(X\right)$ is homeomorphic to $Ends\left(Y\right)$. Thus we propose an alternative definition of $Ends\left(G\right)$ which increases the class of spaces on which $G$ can act.