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At the boundary of some hyperbolic polyhedra. (Au bord de certains polyèdres hyperboliques.) (French) Zbl 0820.20043

Summary: The framework of this article is that of the hyperbolic groups and spaces of M. Gromov. It is motivated by the following question: how to differentiate two hyperbolic groups up to quasi-isometry? We illustrate this problem by detailing one of M. Gromov’s examples taken from ‘Asymptotic invariants for infinite groups’ (1993). We describe an infinite family of hyperbolic groups, two by two non quasi-isometric, for which the boundary is Menger’s curve. The method consists of the study of their quasi-conformal structure on the boundary, using a numerical invariant: P. Pansu’s conformal dimension.

MSC:

20F65 Geometric group theory
28A80 Fractals
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M50 General geometric structures on low-dimensional manifolds
53C22 Geodesics in global differential geometry
52B99 Polytopes and polyhedra
28A75 Length, area, volume, other geometric measure theory
54E40 Special maps on metric spaces
51M10 Hyperbolic and elliptic geometries (general) and generalizations
57S25 Groups acting on specific manifolds
53C20 Global Riemannian geometry, including pinching
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