arXiv subjects
Publications and source records attributed to Olivier Glorieux.
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 $\mathrm{PO}(n,2)$.
We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respectively to a highest weight and a simple root.
In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate
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 $\mathfrak a\subset \mathfrak g$ of the Lie algebra of $G$. We show that if $Γ< G$ is a discrete group, and $Γ' \triangleleft Γ$ is a Zariski dense normal subgroup, then the limit cones of $Γ$ and $Γ'$ in $\mathfrak a$ coincide. Moreover, for all linear form $ϕ: \mathfrak a\to \mathbb R$ positive on this limit cone, the critical exponents in the direction of $ϕ$ satisfy $\displaystyle δ_ϕ(Γ') \geq \frac 1 2 δ_ϕ(Γ)$. Eventually, we show that if $Γ'\backslash Γ$ is amenable, these critical exponents coincide.
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.
We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.
In this paper, we study topological properties of the right action by translation of the Weyl Chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing. (1)
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, Guéritaud 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.
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.
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.
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.
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 Teichmüller space.