We construct a geometric, real-analytic parameterization of the Hitchin component of the -character variety of a closed surface . 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 with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of and triangle invariants associated with each component of the complement . We describe identities and relations satisfied by these invariants, and we use the resulting coordinates to parameterize the Hitchin component.
"Parameterizing Hitchin components." Duke Math. J. 163 (15) 2935 - 2975, 1 December 2014. https://doi.org/10.1215/0012794-2838654