Exposés

Marie Abadie:

Sasha Bontemps:

Noyau parfait de l'espace des sous-groupes des groupes de Baumslag-Solitar généralisés

Farid Diaf:

Vector fields on the hyperbolic plane and surfaces in Half-Pipe space

 In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory. In this spirit, I will present a correspondence between vector fields on the hyperbolic plane and surfaces in the so-called Half-Pipe space, which is the dual of Minkowski space. Using this construction, I investigate the problem of extending vector fields on the circle to the hyperbolic plane. Following this, I will focus on two types of extensions: infinitesimal earthquakes and harmonic Lagrangian vector fields. Time permitting, I will also discuss how the properties of these vector fields may be expressed in terms of the properties of the corresponding surfaces in Half-Pipe space, as well as their asymptotic boundaries.

Nastaran Einabadi:

Rotation theory and the fine curve graph on the torus: non-proper parabolic examples

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb in 2022. This hyperbolic metric graph is used to study the homeomorphism group of the surface through its action on the graph.
Since the introduction of this object, many researchers have worked to establish connections between the dynamics of surface homeomorphisms, and the dynamics of their action by isometries on the fine curve graph. These endeavors have led to the discovery of a connection between this subject and the rich area of rotation theory on surfaces. Rotation theory was first developed in the late 1980s by Misiurewicz and Ziemian, and remains an active field of research to this day.
In this talk, we will introduce the fine curve graph, and the classic and generalized rotation sets on the two-dimensional torus. We will then see how, by using the famous approximation by conjugation method of Anosov and Katok, we can construct examples of torus diffeomorphisms that act parabolically and non-properly on the fine curve graph, while admitting a rich family of generalized rotation sets.

Balthazar Fléchelles:

Geometric finiteness, convex projective geometry and relative Anosov representations

In hyperbolic geometry, geometrically finite representations into PO(n,1) enjoy good geometric and dynamical properties, yielding a nice relative stability result due to Bowditch. When a geometrically finite representation has no parabolics, we say it is convex-cocompact, and we get a stability result instead. Anosov representations are a generalization of convex-cocompact representations in PO(n,1) to higher rank Lie groups that admit a similar stability result: any small enough deformation of an Anosov representation remains Anosov. More recently, several generalizations of geometrically finite representations have been defined by relativizing Anosov representations, but most of them seem to satisfy a weaker version of the classical relative stability result of Bowditch. 

In a celebrated work, Danciger, Guéritaud and Kassel established a strong connection between P1-Anosov representations and a convex-cocompactness notion that they define in convex projective geometry. This gives a geometric setup for studying Anosov representations, and helps producing new examples of Anosov representations. In joint work with Mitul Islam and Feng Zhu, we prove a similar result linking geometric finiteness in convex projective geometry with relative Anosov representations in the sense of Zhu-Zimmer and Kapovich-Leeb. As a corollary, we produce a lot of new examples of relative Anosov representations, and we show through an explicit example that the known relative stability results for relative Anosov representations are suboptimal, meaning that a statement with weaker conditions should hold.

Federica Gavazzi:

Virtual Braid Groups, Virtual Artin Groups, and Their Classifying Spaces

In topology, a "braid" is a set of strands in space that wind around each other in a specified direction. Up to isotopy, these form a group called the classical braid group on n strands, denoted by Bn. Virtual braid groups generalize classical braids by introducing additional structure. Classical braid groups are examples of Artin groups—important groups defined by a presentation with generators and relations. Similarly, virtual braid groups are examples of Virtual Artin groups VA, extending the “virtual” concept to all Artin groups, as recently introduced by Bellingeri, Paris, and Thiel. The virtual braid group on n strands, VBn, projects onto the symmetric group Sn via two homomorphisms, each defining a notable subgroup. This naturally extends to the context of virtual Artin groups, which project onto the associated Coxeter group with two distinct homomorphisms, giving rise to two kernels, PVA and KVA.

This talk explores the topology of these two subgroups through associated topological spaces. We construct a topological space Ω  whose fundamental group is isomorphic to PVA, generalizing a former construction (the BEER complex) from pure virtual braids to all Coxeter graphs. We assert that Ω is aspherical when the associated Coxeter graph is of spherical or affine type, making it a classifying space for PVA in these cases. As a consequence, we obtain a classifying space for the pure virtual braid group on n strands.

Viola Giovannini:

Filling Riemann Surfaces by Hyperbolic Schottky Manifolds of Negative Renormalized Volume

Given a hyperbolizable 3-manifold N, the renormalized volume is a real-valued function on the space of convex co-compact hyperbolic structures on the interior of N, which always have infinite hyperbolic volume. The simplest examples of convex co-compact hyperbolic 3-manifolds are the handlebodies, and, given a connected Riemann surface X of genus at least 2, we call Schottky filling of X a handlebody with boundary at infinity X. A question attributed to Maldacena asks whether given a connected Riemann surface X of genus at least two, there exists a Schottky filling of X of negative renormalized volume. We present an upper bound for the renormalized volume in terms of the genus and the hyperbolic curve lengths of a suitable pants decomposition of X, which allows us to positively answer the question of Maldacena for certain classes of Riemann surfaces.

Nolwenn Le Quellec: 

Ushijima coordinates and rigidity of the (simple) orthospectrum

In 1993, Basmajian introduces the orthospectrum: The multiset of lengths of orthogeodesic on a hyperbolic surface with boundary. We ask ourselves "Does the orthospectrum determine up to isometry a hyperbolic surface ?". Masai and McShane proved in 2023 that the anwser is no, but they still gave a result of rigidity for surfaces with only 1 boundary component. On another note, hyperbolic surfaces live in Teichmüller space, which is usually described with Fenchel-Nielsen coordinates. In this talk, we will see how for hyperbolic surfaces with boundary we can use a different set of coordinates to study the rigidity of the orthospectrum and the simple orthospectrum: Ushijima coordinates. 

Basile Morando:

Factorialité des algèbres de von Neumann de groupes localement compacts et de leurs actions

Dans cet exposé, on expliquera d’abord comment l’étude des sous-représentations d’une représentation unitaire $(\pi, \mathcal{H}_{\pi})$ d’un groupe localement compact $G$ se ramène à celle du commutant $\pi(G)’$ de la représentation. Ce commutant motivera la définition de la structure d’algèbre de von Neumann dont il est muni. On interprétera alors la \emph{factorialité} (la trivialité du centre) d’une telle algèbre comme une forme affaiblie d’irréductibilité de la représentation $\pi$. 

On s’intéresse alors à la question suivante: quand-est-ce que l’algèbre de Von Neumann associée à la représentation régulière d’un groupe localement compact est-elle factorielle? Si $G$ est discret, on obtient aisément une condition nécessaire et suffisante. A contrario, lorsque $G$ n’est pas discret, la question est largement mal comprise. On donnera quelques exemples, on montrera comment cette question peut être reliée à celle de la factorialité d’algèbres de von Neumann associées à des actions de groupes localement compacts sur d’autres algèbres de von Neumann, et on présentera quelques critères obtenus dans cette situation. 

Alex Moriani:

Polygonal surfaces in pseudo-hyperbolic spaces

A polygonal surface in the pseudo-hyperbolic space H(2,n) is a complete maximal surface bounded by a lightlike polygon in the boundary BH(2,n). We give several characterizations of these surfaces using total curvature or asymptotic flatness. The goal of the talk is to explain some constructions coming from nonpositive curvature geometry to establish the link between being polygonal and having finite total curvature.

Arnaud Nerrière:

Sophie Wright:

Counting subgroups of surface groups

The fundamental group of a hyperbolic surface has an infinite number of rank k subgroups. What does it mean, therefore, to pick a 'random' subgroup of this type? In this talk, I will introduce a method for counting subgroups and discuss how counting allows us to study the properties of a random subgroup and its associated cover.

Ivan Yakovlev:

Metric ribbon graphs

Metric ribbon graphs are graphs embedded into surfaces whose edges are equipped with some positive lengths. One naturally comes to the problem of their enumeration when studying random square-tiled surfaces, for example. I will show how bijections from the theory of combinatorial maps can help obtain explicit enumerative formulas for these objects. If time permits, I will also present a byproduct of one of the constructions: a curious family of triangulations of a product of simplices.