Planar boundaries and parabolic subgroups

G. Christopher Hruska, Genevieve S. Walsh

Published 2020-08-17Version 1

We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that, if there are parabolic inseparable cut pairs in $\partial(G,\mathcal{P})$, the group $G$ splits over a finite group. We use this to prove that when $G$ is one ended, and $(G, \mathcal{P})$ is a relatively hyperbolic pair with connected planar boundary with no cut points, then 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.