An introduction to pressure metrics for higher Teichmüller spaces

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.

Simple length rigidity for Hitchin representations

“…A result of Wolpert [34] implies that the restriction of the pressure metric to the Fuchsian locus is a multiple of the classical Weil-Petersson metric. (See [7] for a survey of this theory.) Theorem 8.2.…”

The calculus of thermodynamical formalism

“…More precisely, we construct a Riemannian metric on the manifold of normalized potentials, which corresponds equivalently to a Riemannian metric on the quotient space Q = X (Ω)/C, and relates in various ways to dynamical quantities. After a conformal rescaling by the metric entropy, this metric is very closely related to the metric defined by McMullen [McM08] (see also [BCS15] and references therein).…”

Counting Geodesics, Teichmüller Space, and Random Hyperbolic Surfaces

“…A Riemann surface 𝑅 is prescribed by a covering of a topological space by an atlas of charts {(𝑈 𝛼 , 𝑧 𝛼 )} with 𝑧 𝛼 maps to the complex plane ℂ such that the overlap maps 𝑧 𝛽 ∘ 𝑧 −1 𝛼 are biholomorphic. A Riemannian metric for a surface is prescribed by specifying positive coefficients {(𝑈 𝛼 , 𝑔 𝛼 )} such that 𝑔 𝛼 = 𝑔 𝛽 | 𝑑𝑧 𝛽 𝑑𝑧 𝛼 | 2 . Alternately, a hyperbolic metric can be specified by special charts-{(𝑈 𝛼 , 𝑤 𝛼 )} with 𝑤 𝛼 maps to the hyperbolic plane ℍ with 𝑤 𝛽 ∘ 𝑤 −1 𝛼 local isometries.…”