arXiv subjects

Ludovic Marquis

Publications and source records attributed to Ludovic Marquis.

At least 19 recordsLinked to original sources

arXiv2026

On 4-dimensional convex projective domains invariant by a lattice of $\mathrm{SL}_2 (\mathbb{R})$

This paper is a sequel to the erratum by the authors to a paper by Crampon and Marquis (see arXiv:1202.5442). The main result of the Erratum was relating several notions of geometrical finiteness in round convex projective geometry and we prove here that our series of implications was sharp, by providing counterexamples to the implications that were not established. Our counterexamples are 4-dimensional convex domains $Ω$ acted on by $ρ(Γ)$ where $Γ$ is a lattice of $\mathrm{SL}_2 (\mathbb R)$ and $ρ$ is the irreducible representation of $\mathrm{SL}_2 (\mathbb R)$ of dimension $5$. We give a description of all $ρ(Γ)$-invariant convex domains, and in particular we construct one which is "close enough" to the convex hull $\mathcal C$ of the limit set of $ρ(Γ)$ so that the Hilbert volume $\mathrm{Vol}_{Ω/Γ}(\mathcal C/Γ)$ of the convex core is infinite. We include an appendix with a smoothing procedure in the spirit of Cooper, Long and Tillman (arXiv:1511.06206) and Danciger, Guéritaud and Kassel (arXiv:1704.08711).

math.GT↗

arXiv2026

Convex cocompact groups with three-dimensional limit sets

We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and outputs a convex cocompact right-angled reflection group acting on real hyperbolic n-space whose nerve is precisely the Przytycki-Świątkowski subdivision of L. Moreover, the output reflection group is a thin subgroup of an n-dimensional cocompact arithmetic hyperbolic lattice. This answers affirmatively a question of M. Kapovich concerning the existence of a convex cocompact group acting on some real hyperbolic space with limit set a Čech cohomology sphere other than the standard sphere.

math.GR↗

arXiv2026

Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere

We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by $50$ reflections of $\mathbb{H}^4$, and the other by a rotation of order $21$ and a reflection of $\mathbb{H}^6$. For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.

math.GT↗

arXiv2025

Zariski-Closures of Linear Reflection Groups

We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of $\mathrm{SL}_n(\mathbb{Z})$ that are not finitely presented for all $n \geq 6$.

math.GT↗

arXiv2024

Finitude géométrique en géométrie de Hilbert + an erratum/addendum

The paper is divided in 2 parts. The first part is the original paper of the second and third authors arXiv:1202.5442v2. The second part is an erratum/addendum written in english and concatenated at the end of the former paper. In the erratum/addentum, we amend Theorems 1.3 and 1.11 of arXiv:1202.5442v2: Finitude géométrique en géométrie de Hilbert. We seize the opportunity to show that in round Hilbert geometry, geometrical finiteness (gf) is equivalent to cusp-uniform action and to fill some small gaps that appear in two other proofs of arXiv:1202.5442v2.

math.GT↗

arXiv2024

Convex cocompactness for Coxeter groups

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 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 Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711.

math.GR↗

arXiv2022

Deformation spaces of Coxeter truncation polytopes

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 $Γ$ of the group of projective transformations so that the $Γ$-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 $Ω$ of the $Γ$-orbit of $P$ is convex and $Γ$ acts properly discontinuously on $Ω$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus Ω$ 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.

math.GT↗

arXiv2022

Discrete Coxeter groups

Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one uses Coxeter groups to construct interesting examples of discrete subgroups of Lie groups.

math.GT↗

arXiv2021

A small closed convex projective 4-manifold via Dehn filling

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.

math.GT↗

arXiv2020

Properly convex bending of hyperbolic manifolds

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $Σ$ 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 $d\geq 3$ 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.

math.GT↗

arXiv2018

Convex projective generalized Dehn filling

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.

math.GT↗

arXiv2018

Anti-de Sitter strictly GHC-regular groups which are not lattices

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 [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] and 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.

math.GT↗

arXiv2017

Surface groups of diffeomorphisms of the interval

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, is topologically transitive, and such that only countably many points of the interval $(0,1)$ have non-trivial stabiliser.

math.GT↗

arXiv2016

Deformations of convex real projective manifolds and orbifolds

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups.

math.GT↗

arXiv2016

Entropy rigidity of Hilbert and Riemannian metrics

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 Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang.

math.DG↗

arXiv2015

Coxeter group in Hilbert geometry

A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ 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 $Γ$, find the maximal $Γ$-invariant convex, when there is a unique $Γ$-invariant convex, when the convex $Ω$ is strictly convex, when we can find a $Γ$-invariant convex $Ω'$ which is strictly convex.

math.GT↗

arXiv2014

Around groups in Hilbert Geometry

This is survey about action of group on Hilbert geometry. It will be a chapter of the "Handbook of Hilbert geometry" edited by G. Besson, M. Troyanov and A. Papadopoulos.

math.GT↗

arXiv2013

Le flot géodésique des quotients geometriquement finis des géométries de Hilbert

We study the geodesic flow of geometrically finite quotients $Ω/Γ$ 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 $Ω$ and the group $Γ$. 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.

math.DS↗