Hitchin characters and geodesic laminations

TL;DR

This paper develops a new parametrization of the Hitchin component for closed surfaces, extending Thurston's shear coordinates from Teichmüller space to higher rank Lie groups, with several applications.

Contribution

It introduces a novel parametrization of Hitchin components using geodesic laminations, generalizing Thurston's shear coordinates to higher dimensions.

Findings

New parametrization of Hitchin components for closed surfaces.

Extension of Thurston's shear coordinates to higher rank groups.

Applications demonstrating the utility of the new parametrization.

Abstract

For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the surface. This is a natural extension of Thurston's parametrization of the Teichmueller space of S by shear coordinates associated to a maximal geodesic lamination, corresponding to the case n=2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.