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.