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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Carathéodory's metrics on Teichmüller spaces and L-shaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497–535.
One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each Teichmüller space of a closed surface of genus g≥2. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential.
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- Award ID(s):
- 1800742
- NSF-PAR ID:
- 10104460
- Date Published:
- Journal Name:
- Duke Math. J
- Volume:
- 167
- Issue:
- 3
- Page Range / eLocation ID:
- 497–535
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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