Duke Mathematical Journal

Parameterizing Hitchin components

Francis Bonahon and Guillaume Dreyer

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We construct a geometric, real-analytic parameterization of the Hitchin component Hitn(S) of the PSLn(R)-character variety RPSLn(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 λS with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of λ and triangle invariants associated with each component of the complement Sλ. We describe identities and relations satisfied by these invariants, and we use the resulting coordinates to parameterize the Hitchin component.

Article information

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

First available in Project Euclid: 1 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Hitchin component Hitchin representation Anosov representation positive representation geodesic lamination shear invariant


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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