Parametrizing Hitchin components
speaker: Guillame Dreyer (Department of Mathematics, University of Notre Dame)abstract: Let \(S\) be a closed, connected, oriented surface of negative Euler characteristic. For \(n\geq 3\), Hitchin components \(\mathrm{Hit}_n(S)\) are components of the \(\mathrm{PSL}_n(\mathbb{R})\)--character variety \(\mathcal{R}_{\mathrm{PSL}_n(\mathbb{R})}(S)\) that correspond to Teichmüller components \(\mathcal{T}(S)\) in the case where \(n=2\). Over the recent years, groundbreaking work has revealed fundamental geometric, dynamical and algebraic properties for the representations in \(\mathrm{Hit}_n(S)\). In particular, these Hitchin representations turn out to share many features with the classic Fuchsian representations.
In a joint work with Francis Bonahon, we construct a geometric, real analytic parametrization of the Hitchin components \(\mathrm{Hit}_n(S)\) that it is based on topological choices only. The construction strongly relies on two independent approaches to studying Hitchin representations: the dynamical approach of Anosov representation, introduced by F. Labourie; and the algebraic-combinatorial approach of Positive representation, developed by V. Fock and A. Goncharov. In essence, given a maximal geodesic lamination \(\lambda\) in \(S\), our parametrization is an extension of Thurston's shear coordinates along the leaves of \(\lambda\) on the Teichmüller space \(\mathcal{T}(S)\), combined with Fock-Goncharov's coordinates on the moduli space of positive framed local systems of a punctured surface.
timetable:
Thu 12 Jun, 16:00 - 17:00, Aula Dini
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