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

Published online by Cambridge University Press:  04 December 2017

GEORGIOS DASKALOPOULOS
Affiliation:
Mathematics Department, Box 1917, Brown University, Providence, RI 02912, USA email daskal@math.brown.edu
CHIKAKO MESE
Affiliation:
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA email cmese@math.jhu.edu
ANDREW SANDERS
Affiliation:
Mathematisches Institut, Universitat Heidelberg, Mathematikon, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany email asanders@mathi.uni-heidelberg.de
ALINA VDOVINA
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK email alina.vdovina@newcastle.ac.uk

Abstract

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.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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