Une surface projective convexe est le quotient d’un ouvert proprement convexe de l’espace projectif réel par un sous-groupe discret de . 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 n’est pas un triangle alors est strictement convexe, à bord et qu’une surface projective convexe 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 of the projective real space by a discrete subgroup of . 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 is not a triangle, then is strictly convex, with boundary and that a convex projective surface is of finite volume if and only if the dual surface is of finite volume.
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
@article{AIF_2012__62_1_325_0, author = {Marquis, Ludovic}, title = {Surface {Projective} {Convexe} de volume fini}, journal = {Annales de l'Institut Fourier}, pages = {325--392}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {62}, number = {1}, year = {2012}, doi = {10.5802/aif.2707}, zbl = {1254.57015}, mrnumber = {2986273}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/aif.2707/} }
TY - JOUR AU - Marquis, Ludovic TI - Surface Projective Convexe de volume fini JO - Annales de l'Institut Fourier PY - 2012 SP - 325 EP - 392 VL - 62 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2707/ DO - 10.5802/aif.2707 LA - fr ID - AIF_2012__62_1_325_0 ER -
Marquis, Ludovic. Surface Projective Convexe de volume fini. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 325-392. doi : 10.5802/aif.2707. http://www.numdam.org/articles/10.5802/aif.2707/
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