On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension

Abstract

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary ${\partial }_{\infty }G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on ${\mathbb{H}}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on ${\mathbb{H}}^3$, and every subgroup in $\mathcal{H}$ is virtually ${\mathbb{Z}}^2$.

19 pages, no figure