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Metric conformal structures and hyperbolic dimension. (English) Zbl 1165.20035
Suppose \((X,d)\) is a \(\text{CAT}(-1)\) space with the Gromov product \((x\mid y)_a:=\tfrac 12(d(a,x)+ d(a,y)-d(x,y))\), where \(a,x,y\in X\). M. Bourdon showed [in Enseign. Math., II. Sér. 41, No. 1-2, 63-102 (1995; Zbl 0871.58069)] that for any \(\varepsilon\in(0,1]\), the formula \(d_a(x,y):=e^{\varepsilon(x\mid y)_a}\) gives a metric on \(\partial X\). In the book of M. R. Bridson and A. Haefliger [Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319. Berlin: Springer (1999; Zbl 0988.53001)] it is written that “However one cannot construct visual metrics on the boundary of arbitrary hyperbolic spaces in such direct manner”.
The paper under review proves that such a direct construction is nevertheless possible for hyperbolic complexes, i.e., simplicial complexes whose 1-skeleton is a hyperbolic graph of uniformly bounded valence. For example, a Cayley graph of a hyperbolic group can be viewed as such. Let \(X\) be a hyperbolic complex, then the author proves that with such a metric \(d\), the \(\text{Isom}(X)\)-action on \(\partial X\) is bi-Lipschitz, Möbius, symmetric and conformal. Then he defines (by analogy with the standard stereographic projection on \(\mathbb{R}^n\)) a stereographic projection \(d_b\colon(\partial X\setminus\{b\}\times\partial X\setminus\{b\})\to[0,\infty)\), where \(b\in\partial X\) and shows that it is a metric conformally equivalent to the metric \(d\). There is also introduced a notion of hyperbolic dimension for hyperbolic spaces with group actions.

MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20F69 Asymptotic properties of groups
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
54E45 Compact (locally compact) metric spaces
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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