Metric conformal structures and hyperbolic dimension

Mineyev, Igor

doi:10.1090/S1088-4173-07-00165-8

Metric conformal structures and hyperbolic dimension
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by Igor Mineyev

Conform. Geom. Dyn. 11 (2007), 137-163

DOI: https://doi.org/10.1090/S1088-4173-07-00165-8

Published electronically: September 12, 2007

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Abstract:

For any hyperbolic complex $X$ and $a\in X$ we construct a visual metric $\check {d}=\check {d}_a$ on $\partial X$ that makes the $\operatorname {Isom}(X)$-action on $\partial X$ bi-Lipschitz, Möbius, symmetric and conformal.

We define a stereographic projection of $\check {d}_a$ and show that it is a metric conformally equivalent to $\check {d}_a$.

We also introduce a notion of hyperbolic dimension for hyperbolic spaces with group actions. Problems related to hyperbolic dimension are discussed.

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