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Parameterizing Hitchin components. (English) Zbl 1326.32023

Summary: We construct a geometric, real-analytic parameterization of the Hitchin component \(\mathrm{Hit}_n(S)\) of the \(\mathrm{PSL}_n(R)\)-character variety \(\mathcal{R}_{{\mathrm{PSL}}_n(\mathbb R)}(S)\) 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.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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