Geodesic
Supra-maximal representations from fundamental groups of punctured spheres to  PSL (2,ℝ)
[Représentations supra-maximales des groupes fondamentaux de sphères épointées dans PSL (2,ℝ).]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1305-1329.

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We study a particular class of representations from the fundamental groups of punctured spheres Σ0,n to the group PSL(2,ℝ), which we call supra-maximal. Though most of them are Zariski dense, we show that supra-maximal representations are totally non hyperbolic, in the sense that every simple closed curve is mapped to an elliptic or parabolic element. They are also shown to be geometrizable (apart from the reducible 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 varieties, the components of supra-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 Fubini-Study form that we compute explicitly. This generalizes a result of Benedetto-Goldman [3] for the sphere minus four points.

Nous étudions une classe particulière de représentations du groupe fondamental des sphères épointées Σ0,n dans le groupe PSL (2,ℝ), que nous appelons supra-maximales. Bien qu'elles soient pour la plupart Zariski denses, nous montrons qu'elles sont totalement non hyperboliques, au sens où l'image de toute courbe fermée simple est elliptique ou parabolique. Nous montrons aussi qu'elles sont géométrisables (hormis celles qui sont réductibles) en un sens très fort : pour tout élément de l'espace de Teichmüller 𝒯0,n, il existe une unique application équivariante holomorphe à valeurs dans le demi-plan inférieur ℍ-. Nous montrons également que les représentations supra-maximales forment des composantes compactes des variétés de caractère relatives. Munies de la structure symplectique de Atiyah-Bott-Goldman, ces composantes sont symplectomorphes à l'espace projectif complexe de dimension n-3 muni d'un multiple de la forme de Fubini-Study que nous calculons explicitement. Cela généralise un résultat de Benedetto-Goldman pour la sphère à quatre trous.

Publié le :
DOI : 10.24033/asens.2410 
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     author = {Deroin, Bertrand and Tholozan, Nicolas},
     title = {Supra-maximal representations  from fundamental groups  of punctured spheres to~$\mathrm {PSL} (2,\mathbb {R})$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1305--1329},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {5},
     year = {2019},
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Deroin, Bertrand; Tholozan, Nicolas. Supra-maximal representations  from fundamental groups  of punctured spheres to $\mathrm {PSL} (2,\mathbb {R})$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1305-1329. doi : 10.24033/asens.2410. http://geodesic.mathdoc.fr/articles/10.24033/asens.2410/

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