Igor Mineyev

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Metric conformal structures in hyperbolic groups

For any (uniformly locally finite) hyperbolic complex  X  I will present a construction of a visual metric  $\check{d}$  on the ideal boundary of  X  that makes the  Isom(X)-action on the boundary bi-Lipschitz, Möbius, and conformal. All this in particular applies to groups that are hyperbolic in the sense of Gromov. The definition of  $\check{d}$  is based on a “nice” metric  $\hat{d}$  on a hyperbolic group that was constructed in a joint paper with G. Yu. On a very informal level, the averaging process involved in the construction of these metrics can be thought to resemble the Ricci flow on manifolds, though we work with cell complexes and in the absence of any Riemannian structure. The visual metric is of interest in particular because of its relation to the Cannon's conjecture that is a group theoretic analog of Thurston's hyperbolization conjecture for  3-manifolds.

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