Hyperbolic groups with planar boundaries

Abstract

We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than $2$ or if it acts geometrically on a CAT(0) cube complex.

This new version contains a proof of the quasi-isometric rigidity of the class of convex-cocompact Kleinian groups. The structure of the text has been reorganized

References in corpus (2)

• Strong accessibility for finitely presented groups
• Local rigidity for hyperbolic groups with Sierpiński carpet boundaries