Rencontre "Groupes et Géométrie" à la maison du Kleebach

Titres et résumés

• Caterina Campagnolo: (Foliated) simplicial volume of manifolds foliated by circles.
Take an aspherical manifold. If its simplicial volume vanishes, what about its Euler characteristic? Asked by Gromov in the 90s, this still open question has given rise to several variants of the simplicial volume, as well as to many attempts to understand their interactions. We will define the foliated simplicial volume and will explain how it can help approaching Gromov's question. We will then present the case of manifolds foliated by circles, for which we obtain an answer to the question. This is joint work with Diego Corro.


• Colin Davalo : Description of domains of discontinuity for some representations of surface groups.
For a representation of a discrete group into a Lie group G an interesting problem is to find a domain of discontinuity, i.e an open domain of an homogeneous space on which the representation acts properly and cocompactly. Finding such a domain enables to associate to the representation the quotient of this domain, which is a compact manifold together with a geometric structure in the sense of Klein. Some special examples of representations can be constructed by taking a discrete representation of the fundamental group of a surface S into PSL(2,R) and compose it with a representation of PSL(2,R) into an other Lie group G. Our goal will be to describe the quotients of some domains of discontinuity associated to these kind of representations, and in particular to prove that in some cases the quotient fibers over the surface S.


• Bruno Duchesne : The infinite dimensional hyperbolic space as universal geometric object.
The hyperbolic plane is a well-known metric space and such spaces exist in any dimension, even infinite. Following Monod and Py, we will introduce hyperbolic kernels that can be great tools to provide embeddings in this space and show somehow its « universality ». We will relate these objects to the space of shapes of Euclidean bodies (after Debin and Fillastre), which embeds in this infinite dimensional hyperbolic space and discuss the question of its completion.




• Anthony Genevois : How to think about wreath products geometrically.
After a quick introduction to wreath products of groups, I will describe a few key ideas that can be used to understand their large-scale geometry and illustrate them by proving some elementary statements. This is based on a common work with R. Tesserra.


• Maxime Gheysens : Arens--Eells Spaces.
Any metric space isometrically embeds into a canonical Banach space, called its Arens--Eells or Lipschitz-free space. This functor has been extensively, but not exhaustively, studied by Banach space theorists. In this talk I will survey some important results about this tool as well as some examples related to Euclidean spaces, ultrametric spaces and trees.


• Thomas Haettel : Actions of groups on injective metric spaces and Helly graphs.
Metric spaces where each collection of pairwise intersecting balls has a non-empty global intersection are called injective metric spaces. Their graph analogue are called Helly graphs. Such spaces enjoy various properties typical of nonpositive curvature. We will review examples of groups having nice actions on such spaces, including hyperbolic groups, lattices in Lie groups, mapping class groups and Artin groups. We will also review consequences of such actions.


• Hermès Lajoinie : Linear escape for random walks on Gromov-hyperbolic spaces.
The goal is to study linear escape of random walks on general hyperbolic spaces. We will show that the distance of non-elementary walks to a basepoint grows linearly, with a probability exponentially close to 1. This results applies without any moment condition on the walk or further assumption on the space like properness. This is based on a paper due to S.Gouëzel.


• Jean Lécureux : Central limit theorems and contact graphs for cubical complexes.
Let G be a group acting on a CAT(0) cube complex. We prove that a random walk on G satisfies a Central Limit Theorem, under mild assumptions. Our main tool is the contact graph of the complex : we compare the random walk in the complex to the random walk in the graph in order to obtain useful estimates. In the course of the proof, weidentify the boundary of the contact graph to a subset of the Roller boundary of the complex. This is joint work with Talia Fernós and Frédéric Mathéus.




• Corentin Le Bars : Rank one isometries and convergence to the boundary in CAT(0) spaces.
Let G be a countable group acting by isometries on a proper CAT(0) space X. A natural question is to ask whether or not the pushforward of a given random walk on G converges in X to a point of its visual boundary. Karlsson and Margulis have proven that the random walk converges when we assume that the drift is positive. In the talk we propose to demonstrate that it is still the case if we replace the hypothesis on the drift by the existence of a rank one isometry g in G.


• Claudio Llosa Isenrich : Dehn functions of coabelian subgroups of direct products of groups.
The Dehn function of a finitely presented group provides a quantitative measure of its finite presentability by measuring the numbers of relations required to detect if a word in the generators represents the trivial element in the group. Since nontrivial coabelian subgroups of direct products of free groups always have interesting non-finiteness properties, it is natural to ask if they also have interesting Dehn functions. I will present recent progress on this topic. We show that under the assumption that the group arises as kernel of a homomorphism onto an abelian group of sufficiently small rank its Dehn function has to be quadratic. This generalises work of Carter and Forester on the Dehn functions of the Stallings--Bieri groups. This is joint work with Robert Kropholler.


• Paul Meunier : Are Haar measures fixed points?
Let G be a locally compact group. A Haar measure on G is a positive left-invariant Radon measure. By an old and fundamental theorem due to Weyl, every locally compact group admits Haar measures, and they are all equal up to a multiplicative constant. A way of formulating this result is that the action of G by left translation on the space of Radon measures always fixes exactly one line. Thus it resembles a fixed-point formulation. However, the original proof is not dynamical. Very early, mathematicians have sought a dynamical proof of the existence of Haar measures, but such proofs have been found only in particular cases. In the first part of the talk, I will sketch a short and oriented survey of such proofs. In the second part, I will present an on-going work with Bruno Duchesne in order to reach a dynamical proof for the general case.


• Francisco Nicolás Cardona : Kähler groups admitting a surface group as a normal subgroup.
Kähler groups form an interesting class of finitely presented groups. Given a Kähler group, what can be said about this group just by considering the action by conjugation on a normal subgroup ? I will present some results of my PhD thesis where I studied the case when the normal subgroup is a surface group and the conjugation action preserves the conjugacy class of a simple closed curve in the underlying surface of the surface group. With this purpose, I will recall the construction of the Bass-Serre tree associated to the decomposition of the surface group as an amalgamated product or HNN extension. I will explain how to extend the action of the surface group on this tree to the Kähler group under the previous situation. Finally I will give an answer to the beginning question (for this particular situation) and some applications of this result.


• Pierre Py : J'essaierai d'expliquer (au moins les grandes lignes) d'un article récent de Stover et Toledo (voir arXiv:2108.12404) qui établit l'existence en toutes dimensions de variétés kählériennes compactes à courbure négative n'ayant pas le type d'homotopie de variétés localement symétriques. L'analogue riemmannien de ce résultat était déjà connu depuis un travail célèbre de Gromov et Thurston dans les années 80.



• Jean Raimbault : Invariant subgroups in the Neretin groups, after Tianyi Zheng.
I will quickly explain what an invariant random subgroup is and give some background in the context of familair (Lie and discrete) groups before introducing the Neretin groups and surveying Zheng's proof that they admit no invariant random subgroups beyond the obvious ones.


• Gonzalo Ruiz Stolowicz : Representations on hyperbolic spaces.
We are going to introduce some tools for the study of the representations of a group on the infinite dimensional hyperbolic space. Specially results around the representations of the group of isometries of a tree and the group of isometries of the (complex) hyperbolic space.


• Mireille Soergel : Systolic complexes and Garside groups.
Systolicity is a combinatorial form of non positive curvature for simplicial complexes. On the other hand every Garside group has a presentation for which the Cayley 2-complex is simplicial. I will give a classification of the Garside groups for which this presentation leads to a systolic complex.


• David Zheng-Xu : Coxeter groups acting on the infinite dimensional hyperbolic space.
Irreducible Coxeter groups are divided into three categories: spherical, affine and hyperbolic. They are discrete groups of reflections acting respectively on the unit sphere S^(n-1) of R^n, a Euclidean n-space or hyperbolic n-space. However, all the classical theory of Coxeter groups and their classification deal with finitely generated groups (which act on finite dimensional spaces). In the spherical case, we can easily produce infinitely generated Coxeter groups acting on an infinite dimensional sphere by "extending the Coxeter diagrams". The question of the existence of infinitely generated hyperbolic Coxeter groups arises when looking for an analog of cocompact lattices in the group of isometries of the infinite dimensional hyperbolic space. In this talk, I will present a construction of Coxeter groups acting irreducibly on the infinite dimensional hyeprbolic space.