An introduction to pressure metrics on higher Teichm\"uller spaces

TL;DR

This paper surveys the use of thermodynamic formalism to define pressure metrics on higher Teichmüller spaces, extending classical concepts to Anosov representations in semi-simple Lie groups.

Contribution

It introduces a unified approach to constructing pressure metrics on higher Teichmüller spaces, including Hitchin components and general Anosov representations.

Findings

Pressure metrics generalize classical Teichmüller space metrics.

The paper discusses open problems in the field.

Connections between thermodynamic formalism and geometric structures.

Abstract

We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichm\"uller spaces. Our higher Teichm\"uller 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 Teichm\"uller 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-Sambarino "\emph{The pressure metric for Anosov representations}" and discusses questions and open problems which arise.