arXiv subjects
Publications and source records attributed to Ludovic Marquis.
We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set $Ø$ and any point $x \in Ø$, any discrete group generated by a finite number of automorphisms of $Ø$, which displace $x$ at a distance less than $\varepsilon_n$, is virtually nilpotent.
A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then $Ω$ is strictly convex, with $\Cc^1$ boundary and that a convex projective surface $S$ is of finite volume if and only if the dual surface is of finite volume.
We build examples of properly convex projective manifold $Ω/ Γ$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$ which are not lattice, nor Schottky groups. Moreover, the open properly convex set $Ω$ is strictly-convex, even Gromov-hyperbolic. Nous construisons des exemples de variétés projectives $Ω/ Γ$ proprement convexes de volume fini, non hyperbolique, non compacte en toute dimension $n \geqslant 2$. Ceci nous permet au passage de construire des groupes discrets Zariski-dense de $\SL_{n+1}(\R)$ qui ne sont ni des réseaux de $\SL_{n+1}(\R)$, ni des groupes de Schottky. De plus, l'ouvert proprement convexe $Ω$ est strictement convexe, même Gromov-hyperbolique.
A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in $(0,\fracπ{2}]$, and the union of $n$ copies of $\R^d$, when the polyhedron has the combinatorics of an \emph{ecimahedron}, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers $n$ and $d$ can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.
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 $β_f(Σ_{g,p})$ of the convex projective structure on the surface $Σ_{g,p}$ of genius $g$ with $p$ punctures is homeomorphic to $\R^{16g-16+6p}$. Finally, we show that $β_f(Σ_{g,p})$ can be identify with a connected component of the space of representation of the fundamental group of $Σ_{g,p}$ in $SL(3,R)$ which keep the parabolic modulo conjugaison.