[Timothée Marquis]    [Publications]    [Seminar]

# Séminaires liés à la théorie des groupes à l'UCLouvain, Chemin du Cyclotron.

## Séminaires à venir

• Lundi 23 mai 2022 au local B335 du Cyclotron à 14h00. (Attention: local inhabituel!)

Mario Klisse (Technical University Delft): A dynamical approach to Hecke operator algebras

 Résumé: Iwahori-Hecke algebras are deformations of the group algebra of Coxeter groups depending on a deformation parameter. They can naturally be represented on the $\ell^2$-space of the corresponding group and thus complete to C*-algebras and von Neumann algebras. The aim of this talk is to introduce and discuss certain topological spaces associated with conntected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. For these, they are closely related to the Hecke operator algebras of the system which allows to apply C*-dynamical methods to the study of these operator algebras. Among other things, we will discuss consequences of this connection for the ideal structure of right-angled Hecke C*-algebras.

• Lundi 20 juin 2022 au local B203 du Cyclotron à 14h00.

Max Carter (The University of Newcastle): Scale groups and their unitary representations

 Résumé: TBA

• Lundi 27 juin 2022 au local B203 du Cyclotron à 14h00.

Ludovic Marquis (Université de Rennes): TBA

 Résumé: TBA

## Séminaires passés

• Lundi 2 mai 2022 au local B335 du Cyclotron à 14h00. (Attention: local inhabituel!)

Piotr Przytycki (McGill University): Groups acting almost freely on $2$-dimensional CAT(0) complexes satisfy the Tits alternative

 Résumé: Let $X$ be a $2$-dimensional complex with piecewise smooth Riemannian metric, finitely many isometry types of cells, that is CAT(0). Let $G$ be a group acting on $X$ with a bound on the cell stabilisers. We will sketch the proof of the Tits alternative saying that $G$ is virtually cyclic, virtually $\mathbf Z^2$ or contains a non-abelian free group. This generalises our earlier work for $X$ a $2$-dimensional systolic complex or a $2$-dimensional Euclidean building. This is joint work with Damian Osajda.

• Lundi 25 avril 2022 au local B203 du Cyclotron à 14h00.

Todor Tsankov (Université Claude Bernard - Lyon 1): Topological dynamics of kaleidoscopic groups

 Résumé: Kaleidoscopic groups are infinite permutation groups recently introduced by Duchesne, Monod, and Wesolek by analogy with a classical construction of Burger and Mozes of subgroups of automorphism groups of regular trees. In contrast with the Burger-Mozes groups, kaleidoscopic groups are never locally compact and they are realized as homeomorphism groups of Wazewski dendrites (tree-like, compact spaces whose branch points are dense). The input for the construction is a finite or infinite permutation group $\Gamma$ and the output is the kaleidoscopic group $K(\Gamma)$. In this talk, I will discuss recent joint work with Gianluca Basso, in which we characterize the metrizability of the universal minimal flow of $K(\Gamma)$ in terms of the original group $\Gamma$. All relevant notions from topological dynamics will be explained.

• Lundi 4 avril 2022 au local B203 du Cyclotron à 14h00.

Marcin Sabok (McGill University): Perfect matchings in hyperfinite graphings

 Résumé: We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. We give several applications of this result. We extend the Lyons-Nazarov theorem by showing that a bipartite Cayley graph admits a factor of iid perfect matching if and only if the group is not isomorphic to the semidirect product of $\mathbf{Z}$ and a finite group of odd order, answering a question of Kechris and Marks in the bipartite case. We also answer a question of Bencs, Hruskova and Toth arising in the study of balanced orientations in graphings. Finally, we show how our results generalize and lead to a simple approach to recent results on measurable circle squaring. Joint work with Matt Bowen and Gabor Kun.

• Lundi 28 mars 2022 au local B203 du Cyclotron à 14h00.

Matthieu Joseph (ENS de Lyon): Allosteric actions of surface groups

 Résumé: In a recent work, I introduced the notion of allosteric actions: a minimal action of a countable group on a compact space, with an ergodic invariant measure, is allosteric if it is topologically free but not essentially free. In the first part of my talk I will explain some properties of allosteric actions, and their links with Invariant Random Subgroups (IRS). In the second part, I will explain a recent result of mine: the fundamental group of a closed hyperbolic surface admits allosteric actions.

• Lundi 21 mars 2022 au local B203 du Cyclotron à 14h00 (exceptionnellement, 2 fois 45 minutes).

Gabor Szabo (KU Leuven): On regularity properties in topological dynamics

 Résumé: Regularity properties have been the primary vehicle for an ongoing exchange of ideas between topological dynamics and C*-algebra theory in the past decade. Whenever one is given an action of a discrete group on a compact space, the crossed product construction yields a C*-algebra whose multiplicative structure encodes the dynamics. If the action is also free and minimal, then the associated crossed product is simple, which makes it interesting to study from the point of view of classification. The recent breakthroughs in the Elliott classification program of simple C*-algebras are driven by the insight that (in stark contrast to the classification of injective factors) one has an unavoidable dichotomy between well-behaved and ill-behaved C*-algebras, the latter of which cannot be classified in any reasonable sense. Since the origins of specific C*-algebraic properties of general crossed products are notoriously hard to pin down at the level of the input data, it is an ongoing challenge to determine in topological dynamical terms when a crossed product C*-algebra of an amenable group action is well-behaved. Over time this has given rise to various dynamical regularity properties that are interesting to study in their own right and even have applications unrelated to C*-algebras. In this talk I shall survey the history of these developments and give a glimpse into the current state-of-the-art, primarily focusing on the regularity property of almost finiteness for actions of amenable groups.

• Lundi 7 mars 2022 au local B203 du Cyclotron à 14h00.

Cyril Houdayer (Université Paris-Saclay): Noncommutative ergodic theory of irreducible lattices in higher rank semisimple algebraic groups

 Résumé: I will survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of irreducible lattices in higher rank semisimple algebraic groups. These results have several applications to ergodic theory, topological dynamics, unitary representation theory and operator algebras. The key novelty in our work is a dynamical dichotomy theorem for boundary structures on (noncommutative) von Neumann algebras. In case of lattices in higher rank simple algebraic groups, I will present a noncommutative analogue of Margulis factor theorem and its relevance regarding Connes rigidity conjecture for group von Neumann algebras.

• Lundi 31 janvier 2022 à 14h00 en ligne (lien dans l'email d'annonce).

Sam Hughes (University of Oxford): Irreducible lattices fibring over the circle

 Résumé: Let $n\geq2$ and let $\Lambda$ be a lattice in a product $\prod_{i=1}^n G_i$ of simple non-compact Lie groups with finite centre. An application of the Margulis Normal Subgroup Theorem implies that if $H^1(\Lambda)$ is non-zero, then $\Gamma$ is reducible. Now, let $\Gamma$ be a lattice in a product of isometry groups of irreducible $\mathrm{CAT}(0)$ spaces $\prod_{i=1}^n X_i$. There are many examples of irreducible $\mathrm{CAT}(0)$ lattices with non-vanishing first cohomology, in this case we can deploy the BNSR invariants and investigate how far these cohomology classes are from a fibration of finite type CW complexes. In this talk we will investigate to what extent the BNSR invariants $\Sigma^m(\Gamma)\subset H^1(\Gamma)$ of $\Gamma$ can be used to determine the reducibility of $\Gamma$.

• Lundi 10 janvier 2022 à 14h00 en ligne (lien dans l'email d'annonce).

Alice Kerr (University of Oxford): Product set growth in mapping class groups

 Résumé: A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.

• Lundi 15 novembre 2021 au local B203 du Cyclotron à 14h00.

Sven Raum (Stockholm University): Simplicity and the ideal intersection property for essential groupoid C*-algebras

 Résumé: Groupoids merge the notion of space and group, yielding a well-adapted framework to study dynamical systems from an algebraic and analytic perspective. For decades, operator algebraists have studied various C*-algebras and von Neumann algebras associated to groupoids, because they provide a bridge to other areas like geometry and dynamics and describe important structural features of operator algebras themselves. In the focus of C*-algebraists, often are étale groupoids, which generalise the notion of discrete groups. To every étale groupoid with locally compact Hausdorff space of units, one can associate an essential groupoid C*-algebra, which is a suitable quotient of the reduced groupoid C*-algebra by an ideal of singular elements. For Hausdorff groupoids, it equals the reduced groupoid C*-algebra. Until recently, it had been an open question to characterise simplicity of such essential groupoid C*-algebras. Even in for Hausdorff groupoids, only partial results were known. In this talk, I will report on joint work with Matthew Kenney, Se-Jin Kim, Xin Li and Dan Ursu, which characterises étale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property. Our characterisation is phrased in terms of what is called essentially confined amenable sections of isotropy groups, a notion that can be checked in concrete cases. This provides a complete solution of the open problem, combining the ideal intersection property with the dynamical requirement of minimality. In particular, it comes as a surprise that non-Hausdorff groupoids fit well into this general picture. Our work extends and unifies previous results among others on C*-simplicty of discrete groups, their topological dynamical systems and groupoids of germs. I will keep this talk accessible for an audience of non-experts, starting with a motivation and explanation of groupoids themselves, before discussing operator algebraic aspects of our work.

• Lundi 4 octobre 2021 au local B203 du Cyclotron à 14h00.

Fathi Ben Aribi (Université Catholique de Louvain): Fuglede-Kadison determinants over free groups and Lehmer’s constants

 Résumé: The Fuglede-Kadison determinant associates a non-negative real number to any equivariant operator acting on the completion of a group algebra. This determinant is technical to define, difficult to compute, and admits connections with the Mahler measure and the hyperbolic volume. In this project, we compute the Fuglede-Kadison determinants of an infinite family of operators over the free groups. To do so, we relate the operators in question with random walks on Cayley graphs, which translates in counting closed paths on regular trees, following works of Bartholdi and Dasbach-Lalin. As a consequence, we give a partial answer to a question of Lück as we establish new upper bounds on Lehmer’s constants for a large family of groups. If time permits, we will mention further applications in constructing topological invariants from representations of braid groups.