Bienvenue sur la page des 1BIOA Chaptal

You will find here my previous research works. Feel free to contact me for any questions at olivier.glrx at gmail dot com.

Olivier Glorieux, Samuel Tapie

We study the critical exponents of discrete subgroups of a higher rank
semi-simple real linear Lie group $G$. Let us fix a Cartan subspace $mathfrak
asubset mathfrak g$ of the Lie algebra of $G$. We show that if $Gamma< G$ is
a discrete group, and $Gamma’ triangleleft Gamma$ is a Zariski dense normal
subgroup, then the limit cones of $Gamma$ and $Gamma’$ in $mathfrak a$
coincide. Moreover, for all linear form $phi : mathfrak ato mathbb R$
positive on this limit cone, the critical exponents in the direction of $phi$
satisfy $displaystyle delta_phi(Gamma’) geq frac 1 2
delta_phi(Gamma)$. Eventually, we show that if $Gamma’backslash Gamma$ is
amenable, these critical exponents coincide.

2020
2020
Olivier Glorieux, Andrew Yarmola

Inspired by classical puzzles in geometry that ask about probabilities of
geometric phenomena, we give an explicit formula for the probability that a
random triangle on a flat torus is homotopically trivial. Our main tool for
this computation involves reducing the problem to new invariant of measurable
sets in the plane that is unchanged under area-preserving affine
transformations. Our result show that this probability is minimized at all
rectangular tori and maximized at the regular hexagonal torus.

Adrien Boulanger, Olivier Glorieux

In this article we consider sequences of random points on non-amenable
coverings of compact manifolds. The sequences are the successive positions of
the trajectories of the Markov process defined recursively by picking a point
uniformly on the Dirichlet domain of the previous one. We prove that the escape
rate is positive in this setting. The main technical point is to show a
non-local isoperimetric inequality from which we get a spectral gap for the
Markov operator. In the case where the covering group is Gromov hyperbolic, we
deduce that almost all trajectories converge in the Gromov boundary and that
the random variables given by the distance from where the process starts
satisfy a central limit theorem.

2020
Olivier Glorieux, Daniel Monclair

The aim of this article is to understand the geometry of limit sets in
pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of
$mathrm{PO}(p,q+1)$ introduced by Danciger, Gu’eritaud and Kassel, called
$mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of
critical exponent and Hausdorff dimension of the limit set. We show that they
are equal and bounded from above by the usual Hausdorff dimension of the limit
set. We also prove a rigidity result in $mathbb{H}^{2,1}=mathrm{ADS}^3$ which
can be understood as a Lorentzian version of a famous Theorem of R. Bowen in
$3$-dimensional hyperbolic geometry.

International Mathematics Research Notices
2019
2019
Olivier Glorieux, Daniel Monclair, Nicolas Tholozan

Limit sets of $mathrm{AdS}$-quasi-Fuchsian groups of $mathrm{PO}(n,2)$ are
always Lipschitz submanifolds. The aim of this article is to show that they are
never $mathcal{C}^1$, except for the case of Fuchsian groups. As a byproduct
we show that $mathrm{AdS}$-quasi-Fuchsian groups that are not Fuchsian are
Zariski dense in $mathrm{PO}(n,2)$.

Olivier Glorieux, Daniel Monclair

Limit sets of $mathrm{AdS}$-quasi-Fuchsian groups of $mathrm{PO}(n,2)$ are
always Lipschitz submanifolds. The aim of this article is to show that they are
never $mathcal{C}^1$, except for the case of Fuchsian groups. As a byproduct
we show that $mathrm{AdS}$-quasi-Fuchsian groups that are not Fuchsian are
Zariski dense in $mathrm{PO}(n,2)$.

2018
2018
Olivier Glorieux

We compare critical exponent for quasi-Fuchsian groups acting on the
hyperbolic 3-space, $mathbb{H}^3$, and on invariant disks embedded in
$mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the
action is Fuchsian and a rigidity theorem for negatively curved surfaces when
the action is quasi-Fuchsian.

Pacific Journal of Math
Nguyen-Thi Dang, Olivier Glorieux

We prove that a sequence of quasi-Fuchsian representations for which the
critical exponent converges to the topological dimension of the boundary of the
group (larger than 2), converges up to subsequence and conjugacy to a totally
geodesic representation.

Ergodic theory and Dynamical System
2018
2017
Olivier Glorieux

For any geodesic current we associated a quasi-metric space. For a subclass
of geodesic currents, called filling, it defines a metric and we study the
critical exponent associated to this space. We show that is is equal to the
exponential growth rate of the intersection function for closed curves.

Olivier Glorieux

We prove that a sequence of quasi-Fuchsian representations for which the
critical exponent converges to the topological dimension of the boundary of the
group (larger than 2), converges up to subsequence and conjugacy to a totally
geodesic representation.

2017
2017
Olivier Glorieux

We propose a definition for the length of closed geodesics in a globally
hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that
the number of closed geodesics of length less than $R$ grows exponentially fast
with $R$ and the exponential growth rate is related to the critical exponent
associated to the two hyperbolic surfaces coming from Mess parametrization. We
get an equivalent of three results for quasi-Fuchsian manifolds in the GHMC
setting : R. Bowen’s rigidity theorem of critical exponent, A. Sanders’
isolation theorem and C. McMullen’s examples lightening the behaviour of this
exponent when the surfaces range over Teichm"uller space.

Geometriae Dedicata