GDR Platon Conference on pseudo-Riemannian geometry
and Anosov representations

• Main page
• Program
• Schedule
• Participants
• Venue & Info
• Registration

Supported by:

Program

There will be four mini-courses and a few talks given by young researchers. Each mini-course will be given by a specialist of the domain.

Michael Wolf will give a colloquium talk.

Andrea Tamburelli, PhD candidate at the Mathematics Department of the University of Luxembourg, will defend his thesis on Thursday, June 14, after the talks of the conference.

Mini-courses

François Guéritaud : Affine actions from Lie group multiplication
The main theme of this series of lectures will be the connection between (i) building equivariant maps for a pair of nearly-equal group actions (say, on a symmetric space with symmetry group $G$), and (ii) building proper actions on affine $n$-space. The basic case is $G=\text{PSL}(2,\mathbb{R})$ and $n=3$. This will include a survey of the main sources of examples of affine quotients, which are still poorly understood in general --- even for free groups.


Olivier Guichard : Characterizations of Anosov representations.
This series of lectures will address the following points:
• intrinsic characterizations of Anosov subgroups (in particular without hyperbolicity assumption)
• some properties of their moduli spaces (mainly openness)
• the associated geometric structures.
Sara Maloni : $H^3$ & $AdS^3$: Quasi-Fuchsian and globally hyperbolic maximal compact manifolds, their convex hulls and a universal point of view.
In this mini-course I will discuss similarities and differences between quasi-Fuchsian manifolds in $H^3$ and globally hyperbolic maximal compact manifolds in $AdS^3$. I will try to underline open questions through all the lectures.
In Lecture 1 I will focus on the hyperbolic case. After revising hyperbolic geometry, I will discuss the theory of Fuchsian and Kleinian groups, which are discrete subgroups of $\text{PSL}(2,\mathbb{R})$ and of $\text{PSL}(2,\mathbb{C})$ respectively, and their actions on $H^2$ and $H^3$. I will mostly focus on quasi-Fuchsian groups. These groups have been studied using complex analytic methods until the 70s when W. Thurston revolutionised the field introducing more topological methods. I will discuss Bers simultaneous uniformisation theorem, whose proof uses complex analytic methods, and then I will discuss some of Thurston's theorems and conjectures about the convex cores of these hyperbolic manifolds.
In Lecture 2 I will focus on the analog theory in anti-de Sitter space, which can be thought as a Lorentzian analog of hyperbolic space. First, I will recall the definition of the anti-de Sitter geometry and then I will define globally hyperbolic maximal compact manifolds, which are an analog of quasi-Fuchisan manifolds. I will then describe the analog of Bers and Thurston's theorems and conjectures, many of which are due to Mess.
Finally, in Lecture 3, if I have time, I will discuss a universal version of these points of views and I will discuss quasi-circles in the boundary of hyperbolic and anti-de Sitter space.


Nicolas Tholozan : Minimal surfaces in Riemannian and pseudo-Riemannian symmetric spaces
The goal of this mini-course is to present some interactions between the study of minimal surfaces and that of Anosov representations.
According to the work of Labourie, given an Anosov representation $\rho$ of a surface group into a semi-simple Lie group $G$, there always exists a $\rho$-equivariant minimal surface in the symmetric space of $G$. The properties of this surface are still poorly understood in general (we don’t know whether it is unique, whether it is embedded, or even immersed). It is sometimes easier to study $\rho$-equivariant menial surfaces into certain \emph{pseudo-Riemannian} $G$-homogeneous spaces.
Though there is (to my knowledge) no general method for proving the existence of such surfaces, we will see that their uniqueness is sometimes easier to prove in a pseudo-Riemannian context. We will explain what these surfaces can tell us about the geometric properties of Anosov representations and the topology of the space of all such representations.

Talks

• Jean-Philippe Burelle: Fundamental domains for positive representations
  
  Positive representations, defined by Fock and Goncharov in 2003, are a class of interesting representations of the fundamental group of a surface with non-empty boundary into $\text{PSL}(n,\mathbb{R})$. Under some additional assumptions, these representations are $B$-Anosov, and so by results of Guichard-Wienhard and Kapovich-Leeb-Porti, they admit cocompact domains of discontinuity in projective space when $n$ is even. I will explain how to build polyhedral fundamental domains for this cocompact action, and for $n=4k+3$ I will explain how a similar construction gives a compact fundamental domain in the sphere $S^{2k+2}$. This is joint work with N. Treib.
  
  
  
• Federica Fanoni: Basmajian-type inequalities and collar lemma for maximal representations
  
  Basmajian's celebrated identity gives a way to compute the length of the boundary of a hyperbolic surface in terms of the lengths of the so-called orthogeodesics (geodesics orthogonal to the boundary at both endpoints). This identity can be generalized to the context of maximal representations -- a class of representations of the fundamental group of a surface that can be seen as a generalization of Teichmüller space. I will describe the classical identity, introduce maximal representations, discuss Basmajian's identity in this setup and show how one can deduce a collar lemma for maximal representations from this result. Joint work with Beatrice Pozzetti.
  
  
  
• Daniel Monclair: Limit sets of Anosov representations
  Anosov representations can be defined as representations in a semi-simple Lie group $G$ having a "nice" limit set in a compact quotient $G/P$. Depending on the pair $(G,P)$, these limit sets can be very different. In rank one (quasi-Fuchsian groups), they are fractal objects. For some examples in higher rank (such as Hitchin representations), they are more regular, and can even be $C^1$ submanifolds of $G/P$. We will discuss some "intermediate" examples for which the limit sets are Lipschitz submanifolds of $G/P$, but not $C^1$ submanifolds. We will also discuss "how fractal" these objects are by looking at a modified version of the Hausdorff dimension. This is joint work with Olivier Glorieux.
  
  
  

Colloquium of Michael Wolf

PhD defense of Andrea Tamburelli