Geometry & Topology, Vol. 9 (2005) Paper no. 7, pages 219--246.

Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

Mario Bonk, Bruce Kleiner

Abstract. Suppose G is a Gromov hyperbolic group, and the boundary at infinity of 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 hyperbolic 3-space.

Keywords. Gromov hyperbolic groups, Cannon's conjecture, quasisymmetric maps

AMS subject classification. Primary: 20F67. Secondary: 30C65.

DOI: 10.2140/gt.2005.9.219

E-print: arXiv:math.GR/0208135

Submitted to GT on 28 July 2003. (Revised 3 April 2004.) Paper accepted 09 December 2004. Paper published 26 January 2005.

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Mario Bonk, Bruce Kleiner
Department of Mathematics, University of Michigan
Ann Arbor, MI, 48109-1109, USA
Email: mbonk@umich.edu, bkleiner@umich.edu

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