Surface Projective Convexe de volume fini

Marquis, Ludovic

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 325-392.

Une surface projective convexe est le quotient d’un ouvert proprement convexe $\Omega$ de l’espace projectif réel ${\mathbb{P}}^2\left(ℝ\right)$ par un sous-groupe discret $\Gamma$ de ${\mathrm{SL}}_3\left(ℝ\right)$. Nous donnons plusieurs caractérisations du fait qu’une surface projective convexe est de volume fini pour la mesure de Busemann. On en déduit que si $\Omega$ n’est pas un triangle alors $\Omega$ est strictement convexe, à bord ${𝒞}^1$ et qu’une surface projective convexe $S$ est de volume fini si et seulement si la surface duale est de volume fini.

A convex projective surface is the quotient of a properly convex open $\Omega$ of the projective real space ${\mathbb{P}}^2\left(ℝ\right)$ by a discrete subgroup $\Gamma$ of ${\mathrm{SL}}_3\left(ℝ\right)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann’s measure. We deduce that, if $\Omega$ is not a triangle, then $\Omega$ is strictly convex, with ${𝒞}^1$ boundary and that a convex projective surface $S$ is of finite volume if and only if the dual surface is of finite volume.

Classification : 57M50, 52C20, 22E40
Mot clés : surface, géométrie de Hilbert, géométrie hyperbolique, réseau, sous-groupes discrets des groupes de Lie
Keywords: Surface, Hilbert’s geometry, Hyperbolic geometry, Lattice, Discrete subgroup of Lie group

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