Relatively hyperbolic groups with planar boundaries

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 $S^2$.