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# Les pavages en géométrie projective de dimension 2 et 3

Abstract : I'm interested in the construction and the study of properly convex projective structure on manifold. A properly convex projective manifold is the quotient of a properly convex open set $\Omega$ by a torsion-free discret group $\Gamma$ of projective transformation which preserve $\Omega$. The leading example is the following: we take for $\Omega$ the projective model of the real hyperbolic space and for $\Gamma$ a lattice in the isometry group $\mathrm{SO}_{n,1}$ of the hyperbolic space. This example is very simple because the open set $\Omega$ is homogeneous and strictly convex. A properly convex open set $\Omega$ is called divisible when there exists a discret subgroup of $\mathrm{PGL}_{n+1}(\mathbb{R})$ which preserve $\Omega$ and such that the quotient $\Omega/_{\Gamma}$ is compact. Divisible convex has been study a lot specially by Yves Benoist and William Goldman. There works show that this structures are natural, and when the open set $\Omega$ is strictly convex, very closed to the hyperbolic structure. In fact, Benoist has shown that $\Omega$ is stricly convex iff the group $\Gamma$ is Gromov-hyperbolic. The construction of a divisible convex $\Omega$ with $\Omega$ not homogeneous is a difficult task. In fact, we know that there exists a divisible convex which is strictly convex and not homogeneous in every dimension. But, the existence of divisible convex $\Omega$ divided by a discret group $\Gamma$ such that $\Omega$ is not strictly convex and not homogeneous and $\Gamma$ irreducible is only known in dimension 3, 4, 5 and 6. And, we know that such example cannot exist in dimension 2. In my PhD, I'm interested to moduli space of this object in dimension 2 and 3. In dimension 3, Vinberg has find a method to construct a divisible convex with a Coxeter group which is analogous to the Poincaré method to construct a lattice in $\mathrm{O}_{n,1}$ with a Coxeter group. One take a polyedra $P$ in the projective real space of dimension 3, one choose one reflection across each face of $P$ such that the product of two reflections which share an edge is conjugate to a rotation of finite order. Then, the group $\Gamma$ generated by this reflection is a Coxeter group, the union $\Omega$ of the image of $P$ is a convex set of the projective space, and $P$ is a fondamental domain for the action of $\Gamma$ on $\Omega$. I've shown that the moduli space $X$ of properly convex projective structure on an infinite family of Coxeter orbifold is the disjoint union of ball of the same dimension. Moreover, the dimension of $X$ and its number of connected component are computable explicitly in function of invariant of the Coxeter group. There is a natural distance on each properly convex open set $\Omega$ called the Hilbert distance. This distance is in fact a Finsler metric and so there is a natural measure on $\Omega$ which is absolutely continuous for the Lebesgue measure. All this objects are invariant under the group of projective transformation which preserve $\Omega$. In dimension 2, I've explored the case where the quotient $\Omega/_{\Gamma}$ is of finite volume. I've shown that a properly convex projective surface is of finite volume if and only if the surface is of finite type and the holonomy of each elementary loop is parabolic. I've computed the moduli space of properly convex projective structure of finite volume on the orientable surface of genus $g$ and with $p$ punctures is homeomorphic to a ball of dimension 16g-16+6p. I've also shown that if the quotient $\Omega/_{\Gamma}$ is of finite volume and $\Omega$ is not a triangle then $\Omega$ is strictly convex and of $C^1$ boundary.
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https://tel.archives-ouvertes.fr/tel-00428902
Contributor : Ludovic Marquis <>
Submitted on : Friday, October 30, 2009 - 10:33:22 AM
Last modification on : Tuesday, November 19, 2019 - 12:54:45 PM
Long-term archiving on: : Thursday, June 17, 2010 - 6:42:56 PM

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• HAL Id : tel-00428902, version 1

### Citation

Ludovic Marquis. Les pavages en géométrie projective de dimension 2 et 3. Mathématiques [math]. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00428902⟩

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