Hitchin characters and geodesic laminations. (English) Zbl 1388.32006

Bonahon, Francis; Dreyer, Guillaume

Acta Math. 218, No. 2, 201-295 (2017).

Authors’ abstract: For a closed surface \(S\), the Hitchin component \(\mathrm{Hit}_n (S)\) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group \(\pi_1(S)\) to the Lie group \(\mathrm{PSL}_n (\mathbb{R})\). We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination \(\lambda\) on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space \(\mathcal{T}(S)\) by shearing coordinates associated with \(\lambda\), corresponding to the case \(n=2\). However, significantly new ideas are needed in this higher-dimensional case. The article concludes with a few applications.

Reviewer: V. V. Chueshev (Kemerovo)

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)

Keywords:

closed surfaces; Hitchin component; geodesic lamination; Thurston’s parametrization of the Teichmüller space