Local cut points and splittings of relatively hyperbolic groups

TL;DR

This paper demonstrates that non-parabolic local cut points in the Bowditch boundary of a relatively hyperbolic group imply the group splits over a 2-ended subgroup, extending known results from hyperbolic groups and classifying boundary types.

Contribution

It generalizes Bowditch's splitting theorem to relatively hyperbolic groups and introduces a broader notion of group ends for boundary classification.

Findings

Non-parabolic local cut points imply group splits over 2-ended subgroups.

Classifies 1-dimensional Bowditch boundaries under certain conditions.

Proposes a new definition of ends for groups based on group actions.

Abstract

In this paper we show that the existence of a non-parabolic local cut point in the Bowditch boundary $\partial(G,\mathbb{P})$ of a relatively hyperbolic group $(G,\mathbb{P})$ 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 $Ends(X)$ is homeomorphic to $Ends(Y)$. Thus we propose an alternate definition of $Ends(G)$ which increases the class of spaces on which $G$ can act.