Surface groups acting on CAT(−1) spaces

Daskalopoulos, Georgios; Mese, Chikako; Sanders, Andrew; Vdovina, Alina

doi:10.1017/etds.2017.103

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Surface groups acting on CAT(−1) spaces

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Daskalopoulos, Georgios
Max Planck Institute for Mathematics, Max Planck Society;

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Vdovina, Alina
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Daskalopoulos, G., Mese, C., Sanders, A., & Vdovina, A. (2019). Surface groups acting on CAT(−1) spaces. Ergodic Theory and Dynamical Systems, 39(7), 1843-1856. doi:10.1017/etds.2017.103.

Cite as: https://hdl.handle.net/21.11116/0000-0004-6595-2

Abstract

Harmonic map theory is used to show that a convex cocompact surface group action on a CAT(-1) metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a CAT(-1) space has Hausdorff dimension ≥1, where the inequality is strict unless the action is
Fuchsian.