Higher Teichmüller Theory for Surface Groups and Shifts of Finite Type

Abstract

The Teichmuller space of Riemann metrics on a compact oriented surface V without boundary comes equipped with a natural Riemannian metric called the Weil–Petersson metric. Bridgeman, Canary, Labourie and Sambarino generalised this to Higher Teichmuller Theory, i.e. representations of π1(V ) in \(\mathrm {SL}(d, \mathbb R)\), and showed that their metric is analytic. In this note we will present a new equivalent definition of the Weil–Petersson metric for Higher Teichmuller Theory and also give a short proof of analyticity. Our approach involves coding π1(V ) in terms of a symbolic dynamical system and the associated thermodynamic formalism.