Surface groups acting on $\text{CAT}(-1)$ spaces

Harmonic map theory is used to show that a convex cocompact surface group action on a $\text{CAT}(-1)$ 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 pro...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2019-07, Vol.39 (7), p.1843-1856
Main Authors: DASKALOPOULOS, GEORGIOS, MESE, CHIKAKO, SANDERS, ANDREW, VDOVINA, ALINA
Format: Article
Language:English
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Summary:Harmonic map theory is used to show that a convex cocompact surface group action on a $\text{CAT}(-1)$ 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 $\text{CAT}(-1)$ space has Hausdorff dimension $\geq 1$ , where the inequality is strict unless the action is Fuchsian.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2017.103