Conformal dimension and Gromov hyperbolic groups with 2–sphere boundary

Abstract

Suppose $G$ is a Gromov hyperbolic group, and ${\partial }_{\infty }G$ is quasisymmetrically homeomorphic to an Ahlfors $Q$–regular metric 2–sphere $Z$ with Ahlfors regular conformal dimension $Q$. Then $G$ acts discretely, cocompactly, and isometrically on ${ℍ}^3$.