Quasicircle boundaries and exotic almost-isometries
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 863-877.

We show that the limit set of an isometric and convex cocompact action of a surface group on a proper geodesic hyperbolic metric space, when equipped with a visual metric, is a Falconer–Marsh (weak) quasicircle. As a consequence, the Hausdorff dimension of such a limit set determines its bi-Lipschitz class. We give applications, including the existence of almost-isometries between periodic negatively curved metrics on 2 that cannot be realized equivariantly.

Nous démontrons que l’ensemble limite d’une action isométrique et convexe co-compacte d’un groupe de surface sur un espace hyperbolique, équipé de sa métrique visuelle canonique, est un (faible) quasi-cercle au sens de Falconer et Marsh. Ceci implique que ces métriques visuelles sur ces ensembles limites sont classifiées, à équivalence bi-Lipschitz près, par leur dimension de Hausdorff. Nous donnons plusiers consequences, y compris l’existence de presque-isométries entre des paires de métriques périodiques sur 2 (malgré le faite qu’en général, il n’existe pas de presque-isométrie équivariante).

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DOI: 10.5802/aif.3098
Classification: 20F67, 51F99
Keywords: Rigidity, quasi-isometry, almost-isometry, bi-Lipschitz map, boundary at infinity, quasi-circle, limit set, Hausdorff dimension.
Mot clés : Rigidité, quasi-isométrie, presque-isométrie, application bi-Lipschitz, bord à l’infini, quasi-cercle, ensemble limite, dimension de Hausdorff.
Lafont, Jean-François 1; Schmidt, Benjamin 2; van Limbeek, Wouter 3

1 Department of Mathematics Ohio State University Columbus, Ohio 43210 (USA)
2 Department of Mathematics Michigan State University East Lansing, MI 48824 (USA)
3 Department of Mathematics University of Michigan Ann Arbor, MI 48109 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lafont, Jean-François; Schmidt, Benjamin; van Limbeek, Wouter. Quasicircle boundaries and exotic almost-isometries. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 863-877. doi : 10.5802/aif.3098. https://aif.centre-mersenne.org/articles/10.5802/aif.3098/

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