Supra-maximal representations from fundamental groups of punctured spheres to $\mathrm{PSL}(2,\mathbb R)$

Bertrand Deroin (1, 2) , Nicolas Tholozan (3, 4)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
3 Uni.lu - Université du Luxembourg
4 LJAD - Laboratoire Jean Alexandre Dieudonné

We study a particular class of representations from the fundamental groups of punctured spheres Σ0,n to the group PSL(2, R) (and their moduli spaces), that we call super-maximal. Super-maximal representations are shown to be totally non hyperbolic, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be geometrizable (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichmüller space T0,n, there is a unique holomorphic equivariant map with values in the lower half-plane H −. In the relative character variety, the components of super-maximal representations are shown to be compact , and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension n−3 equipped with a certain multiple of the Fubiny-Study form that we compute explicitly (this generalizes a result of Benedetto–Goldman [BG99] for the sphere minus four points). Those are the unique compact components in relative character varieties of PSL(2, R). This latter fact will be proved in a companion paper.

Domaines: Mathématiques [math]

Bertrand Deroin, Nicolas Tholozan. Supra-maximal representations from fundamental groups of punctured spheres to $\mathrm{PSL}(2,\mathbb R)$. Annales Scientifiques de l'École Normale Supérieure, 2019, 52 (5), pp.1305-1329. ⟨hal-01328928⟩