Planar boundaries and parabolic subgroups

TL;DR

This paper investigates the boundaries of relatively hyperbolic groups, showing under certain conditions they embed in the sphere and relate to surface groups, supporting the conjecture that such groups are virtually Kleinian.

Contribution

It establishes conditions under which the boundary of a relatively hyperbolic group embeds in a sphere and characterizes the peripheral subgroups as virtually surface groups.

Findings

Boundary embeds in S^2 for rigid groups with no cut points.

Peripheral subgroups are virtually surface groups.

Supports the conjecture that such groups are virtually Kleinian.

Abstract

We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that if $(G,\mathcal{P})$ is a rigid relatively hyperbolic group pair whose boundary embeds in $S^2$, then the action on the boundary extends to a convergence group action on $S^2$. More generally, if the boundary is connected and planar with no cut points, we show that every element of $\mathcal{P}$ is virtually a surface group. This conclusion is consistent with the conjecture that such a group $G$ is virtually Kleinian. We give numerous examples to show the necessity of our assumptions.