Parameterizing Hitchin components

Abstract

We construct a geometric, real-analytic parameterization of the Hitchin component ${\mathrm{Hit}}_n\left(S\right)$ of the ${\mathrm{PSL}}_n\left(\mathbb{R}\right)$-character variety ${\mathcal{R}}_{{\mathrm{PSL}}_n\left(\mathbb{R}\right)}\left(S\right)$ of a closed surface $S$. The approach is explicit and constructive. In essence, our parameterization is an extension of Thurston’s shearing coordinates for the Teichmüller 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 $\lambda \subset S$ 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 we use the resulting coordinates to parameterize the Hitchin component.