Mathematical Research Letters

Volume 30 (2023)

Number 4

Planar boundaries and parabolic subgroups

Pages: 1081 – 1112

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n4.a5

Authors

G. Christopher Hruska (Department of Mathematical Sciences, University of Wisconsin, Milwaukee, Wisc., U.S.A.)

Genevieve S. Walsh (Department of Mathematics, Tufts University, Medford, Massachusetts, U.S.A.)

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.

Received 18 April 2021

Received revised 13 April 2022

Accepted 17 May 2022

Published 3 April 2024