Relatively hyperbolic groups with planar boundaries

TL;DR

This paper proves that certain relatively hyperbolic groups with planar boundaries are virtually Kleinian, advancing understanding of convergence groups on the 2-sphere and their relation to Kleinian groups.

Contribution

It establishes that relatively hyperbolic groups with planar boundaries and specific conditions are virtually Kleinian, confirming a version of Martin and Skora's conjecture.

Findings

Relatively hyperbolic groups with planar boundary are virtually Kleinian.

Results support versions of the Cannon conjecture.

Applications to convergence groups acting on the 2-sphere.

Abstract

In this article, we prove a version of Martin and Skora's conjecture that convergence groups on the $2$-sphere are covered by Kleinian groups. Given a relatively hyperbolic group pair $(G,\mathcal{P})$ with planar boundary and no Sierpinski carpet or cut points in its boundary, and with $G$ one ended and virtually having no $2$-torsion, we show that $G$ is virtually Kleinian. We also give applications to various versions of the Cannon conjecture and to convergence groups acting on $S^2$.