We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic, irreducible Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic irreducible Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in [DGK2].

A convex polytope $P$ in the real projective space with reflections in the facets of $P$ is a Coxeter polytope if the reflections generate a subgroup $\Gamma$ of the group of projective transformations so that the $\Gamma$-translates of the interior of $P$ are mutually disjoint. It follows from work of Vinberg that if $P$ is a Coxeter polytope, then the interior $\Omega$ of the $\Gamma$-orbit of $P$ is convex and $\Gamma$ acts properly discontinuously on $\Omega$. A Coxeter polytope $P$ is $2$-perfect if $P \setminus \Omega$ consists of only some vertices of $P$.

In this paper, we describe the deformation spaces of $2$-perfect Coxeter polytopes $P$ of dimension $d \geqslant 4$ with the same dihedral angles when the underlying polytope of $P$ is a truncation polytope, i.e. a polytope obtained from a simplex by successively truncating vertices. The deformation spaces of Coxeter truncation polytopes of dimension $d = 2$ and $d = 3$ were studied respectively by Goldman and the third author.

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

For $d=4, 5, 6, 7, 8$, we exhibit examples of $\mathrm{AdS}^{d,1}$ strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space $\mathbb{H}^d$, nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot--Mérigot [BM12].

We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion for Coxeter groups built on Danciger--Guéritaud--Kassel [DGK17] aand find examples of Coxeter groups $W$ such that the space of strictly GHC-regular representations of $W$ into $\mathrm{PO}_{d,2}(\mathbb{R})$ up to conjugation is disconnected.

We prove that the group of diffeomorphisms of the interval $[0,1]$ contains surface groups whose action on $(0,1)$ has no global fix point & such that only countably many points of the interval $(0,1)$ have non-trivial stabiliser.

For $d=4, 5, 6$, we exhibit the first examples of complete finite volume hyperbolic $d$-manifolds $M$ with cusps such that infinitely many $d$-orbifolds $M_{m}$ obtained from $M$ by generalized Dehn filling admit properly convex real projective structures. The orbifold fundamental groups of $M_m$ are Gromov-hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d-2}$, hence the images of the developing maps of the projective structures on $M_m$ are new examples of divisible properly convex domains of the projective $d$-space which are not strictly convex, in contrast to the previous examples of Benoist.

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geqslant 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.

In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincaré exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang. The second is in the space of Hilbert metrics and generalizes a result of Crampon.

A theorem of Tits - Vinberg allows to build action of a Coxeter group $\Gamma$ on a properly convex open set $\Omega$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe an hypothesis that make those conditions necessary.

Under this hypothesis, we describe the Zariski closure of $\Gamma$, find the maximal $\Gamma$-invariant convex, when there is a unique $\Gamma$-invariant convex, when the convex $\Omega$ is strictly convex, when we can find a $\Gamma$-invariant convex $\Omega'$ which is strictly convex.

We study the geodesic flow of geometrically finite quotients $\Omega/_{\Gamma}$ of Hilbert geometries, in particular its recurrence properties.
We prove that, under a geometrical assumption on the cusps, the geodesic flow is uniformly hyperbolic. Without this assumption, we provide an example of a quotient whose geodesic flow has a zero Lyapunov exponent.
We make the link between the dynamics of the geodesic flow and some properties of the convex set $\Omega$ and the group $\Gamma$. As a consequence, we get various rigidity results which extend previous results of Benoist and Guichard for compact quotients.
Finally, we study the link between volume entropy and critical exponent; for example, we show that they coincide provided the quotient has finite volume.

We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with $\mathcal{C}^1$ boundary. In Gromov-hyperbolic spaces, geometrical finiteness is defined by a property of the group action on the boundary of the space. We show by constructing an explicit counter-example that this definition has to be strenghtened in order to get equivalent characterizations in terms of the geometry of the quotient orbifold, similar to those obtained by Brian Bowditch in hyperbolic geometry and pinched negative curvature.

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 $\Omega$ and any point $x \in \Omega$, any discrete group generated by a finite number of automorphisms of $\Omega$, which displace $x$ at a distance less than $\varepsilon_n$, is virtually nilpotent.

We build examples of properly convex projective manifold $\Omega/_{\Gamma}$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. This construction gives examples of Zariski-dense discrete subgroups $\Gamma$ of $\textrm{SL}_{n+1}(\mathbb{R})$ which are not lattice of $\textrm{SL}_{n+1}(\mathbb{R})$, nor Schottky group. Moreover, the open properly convex set $\Omega$ build are strictly-convex, even Gromov-hyperbolic. Finally, we give a sufficient condition for gluing of convex projective manifold with geodesic boundary to be a convex projective manifold.

The proof of the main theorem contains a gap (discovered by Sam Ballas). The proof has been repaired and the main theorem improved with Sam in Properly convex bending of hyperbolic manifolds, Groups, Geometry, and Dynamics, volume 14, issue 2, 2020, p. 653–688. The details of the gap can be found in Erratum: Exemples de variétés projectives strictement convexes de volume fini en dimension quelconque, L'Enseignement Mathématique, volume 62, issue 1/2, (2016), p. 3-5.

This article follows the article below in which one characterizes 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 convex projective structures on the surface $\Sigma_{g,p}$ of genius $g$ with $p$ punctures is homeomorphic to $\mathbb{R}^{16g-16+6p}$.
Finally, we show that $\beta_f(\Sigma_{g,p})$ can be identified with a connected component of the space of representations of the fundamental group of $\Sigma_{g,p}$ in $\textrm{SL}_3(\mathbb{R})$ which keep the parabolics modulo conjugaison.

A convex projective surface is the quotient of a properly convex open $\Omega$ of the projective real space $\mathbb{P}^2(\mathbb{R})$ by a discrete subgroup $\Gamma$ of $\textrm{SL}_3(\mathbb{R})$. 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 $\mathcal{C}^1$ boundary and that a convex projective surface $S$ is of finite volume if and only if the dual surface is of finite volume.

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 $(0,\frac{\pi}{2}]$, and the union of $n$ copies of $\mathbb{R}^d$, when the polyhedron has the combinatorics of an 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.