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Title of Journal: Math Ann

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Abbravation: Mathematische Annalen

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Springer Berlin Heidelberg

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DOI

10.1016/0167-6687(87)90001-1

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1432-1807

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On entropy regularity and rigidity for convex rep

Authors: A Sambarino
Publish Date: 2015/04/24
Volume: 364, Issue: 1-2, Pages: 453-483
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Abstract

Given a convex representation rho Gamma rightarrow mathrmPGLdmathbb R of a convex cocompact group Gamma of mathrmIsom +mathbb Hk we find upper bounds for the quantity alpha h rho where h rho is the entropy of rho and alpha is the Hölder exponent of the equivariant map partial infty Gamma rightarrow mathbb Pmathbb Rd We also give rigidity statements when the upper bound is attained This provides an analog of Thurston’s metric for convex cocompact groups of mathrmIsom +mathbb Hk We then prove that if rho pi 1Sigma rightarrow mathrmPSLdmathbb R is in the Hitchin component then alpha h rho le 2/d1 where alpha is the Hölder exponent of Labourie’s equivariant flag curve with equality if and only if rho is FuchsianThe author is extremely thankful to Martin Bridgeman Dick Canary Francois Labourie Alejandro Passeggi and Rafael Potrie for useful discussions He would like to particularly thank Yves Benoist Matias Carrasco and JeanFrançois Quint for discussions that considerably improved the statements of this work and Qiongling Li for pointing out an error on the first version of this paper

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