Duke Mathematical Journal

Parameterizing Hitchin components

Francis Bonahon and Guillaume Dreyer

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Abstract

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

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

Dates
First available in Project Euclid: 1 December 2014

Permanent link to this document
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]

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

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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