Quasicircle boundaries and exotic almost-isometries
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 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).
Received : 2014-10-04
Revised : 2015-02-16
Accepted : 2015-03-26
Published online : 2017-05-31
DOI : https://doi.org/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.
@article{AIF_2017__67_2_863_0,
     author = {Lafont, Jean-Fran\c cois and Schmidt, Benjamin and van Limbeek, Wouter},
     title = {Quasicircle boundaries and exotic almost-isometries},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {863-877},
     doi = {10.5802/aif.3098},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_2_863_0}
}
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/item/AIF_2017__67_2_863_0/

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