## Duke Mathematical Journal

### Parameterizing Hitchin components

#### Abstract

We construct a geometric, real-analytic parameterization of the Hitchin component $\mathrm{Hit}_{n}(S)$ of the $\mathrm{PSL}_{n}(\mathbb {R})$-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.

#### Article information

Source
Duke Math. J., Volume 163, Number 15 (2014), 2935-2975.

Dates
First available in Project Euclid: 1 December 2014

https://projecteuclid.org/euclid.dmj/1417442576

Digital Object Identifier
doi:10.1215/0012794-2838654

Mathematical Reviews number (MathSciNet)
MR3285861

Zentralblatt MATH identifier
1326.32023

Subjects
Primary: 58D57
Secondary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

#### Citation

Bonahon, Francis; Dreyer, Guillaume. Parameterizing Hitchin components. Duke Math. J. 163 (2014), no. 15, 2935--2975. doi:10.1215/0012794-2838654. https://projecteuclid.org/euclid.dmj/1417442576

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