Espace de Modules Marques des Surfaces Projectives Convexes de Volume Fini

TL;DR

This paper characterizes the moduli space of finite volume convex projective structures on punctured surfaces, showing it is homeomorphic to a Euclidean space and relates it to certain representation spaces in SL(3,R).

Contribution

It establishes the topological structure of the moduli space for finite volume convex projective surfaces and links it to representation varieties in SL(3,R).

Findings

Moduli space $eta_f(\Sigma_{g,p})$ is homeomorphic to $ ^{16g-16+6p}$.

The moduli space corresponds to a connected component of representation space in SL(3,R).

Provides a geometric characterization of convex projective structures with finite volume.

Abstract

This article follow the article {http://hal.archives-ouvertes.fr/hal-00361030/fr/} in which the author characterize the fact of being of finite volume for a convex projective surface. We show here that the moduli space $\beta_f(\Sigma_{g,p})$ of the convex projective structure on the surface $\Sigma_{g,p}$ of genius $g$ with $p$ punctures is homeomorphic to $\R^{16g-16+6p}$. Finally, we show that $\beta_f(\Sigma_{g,p})$ can be identify with a connected component of the space of representation of the fundamental group of $\Sigma_{g,p}$ in $SL(3,R)$ which keep the parabolic modulo conjugaison.