Espace des modules marqués des surfaces projectives convexes de volume fini

Marquis, Ludovic¹

¹ Unité de Mathématiques Pures et Appliquées, CNRS UMR 5669, ENS Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07, France

Geometry & topology, Tome 14 (2010) no. 4, pp. 2103-2149.

Résumé

Cet article est la suite de l’article [arXiv :0902.3143] dans lequel l’auteur caractérisait le fait d’être de volume fini pour une surface projective convexe. On montre ici que l’espace des modules ${\beta }_f\left({\Sigma }_{g,p}\right)$ des structures projectives convexes de volume fini sur la surface ${\beta }_f\left({\Sigma }_{g,p}\right)$ de genre $g$ à $p$ pointes est homéomorphe à ${ℝ}^{16g-16+6p}$.

Enfin, on montre que ${\beta }_f\left({\Sigma }_{g,p}\right)$ s’identifie à une composante connexe de l’espace des représentations du groupe fondamental de ${\Sigma }_{g,p}$ dans ${SL}_3\left(ℝ\right)$ qui conservent les paraboliques à conjugaison près.

This article follows the article [arXiv :0902.3143] in which the author characterizes the fact of being of finite volume for a convex projective surface. We show here that the moduli space ${\beta }_f\left({\Sigma }_{g,p}\right)$ of convex projective structures on the surface ${\Sigma }_{g,p}$ of genus $g$ with $p$ punctures is homeomorphic to ${ℝ}^{16g-16+6p}$.

Finally, we show that ${\beta }_f\left({\Sigma }_{g,p}\right)$ can be identified with a connected component of the space of representations of the fundamental group of ${\Sigma }_{g,p}$ in ${SL}_3\left(ℝ\right)$ which keep the parabolics modulo conjugation.

Détail

DOI : 10.2140/gt.2010.14.2103

Keywords: convex projective geometry, surface, moduli space

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