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An introduction to pressure metrics for higher Teichmüller spaces

Published online by Cambridge University Press:  17 March 2017

MARTIN BRIDGEMAN ,
RICHARD CANARY  and
ANDRÉS SAMBARINO
MARTIN BRIDGEMAN
Affiliation:
Boston College, Chestnut Hill, MA 02467, USA
RICHARD CANARY
Affiliation:
University of Michigan, Ann Arbor, MI 41809, USA
ANDRÉS SAMBARINO
Affiliation:
Université Pierre et Marie Curie (Paris VI), 75005 Paris, France
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Abstract

We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.

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Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2001 - 2035
Copyright
© Cambridge University Press, 2017 

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