Parametrizing Hitchin components

TL;DR

This paper introduces an explicit geometric parametrization of Hitchin components for surface groups, extending Thurston's shear coordinates and Fock-Goncharov's positive coordinates, providing a constructive approach to understanding these moduli spaces.

Contribution

It develops a new, explicit parametrization of Hitchin components combining shear and triangle invariants, extending classical and recent coordinate systems.

Findings

Constructs a real analytic parametrization of Hitchin components.

Defines shear and triangle invariants satisfying specific relations.

Provides a coordinate system for the Hitchin component.

Abstract

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a closed surface, combined with Fock-Goncharov's coordinates for the moduli space of positive framed local systems of a punctured surface. More precisely, given a maximal geodesic lamination \lambda in S with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of \lambda; and triangle invariants associated with each component of the complement S-\lambda. We describe identities and relations satisfied by these invariants, and use the resulting coordinates to parametrize the Hitchin component.