| Résumé: |
Reflection groups are images of Coxeter groups under particular representations introduced by Vinberg in the 60's. Symmetry groups of the tiling of the Euclidean or hyperbolic space such that the fundamental tile is a polytope with dihedral angles sub-multiples of $\pi$ and the group is generated by the reflection across the sides of the fundamental tile are examples of reflection groups.
Those representations give actions of Coxeter groups on the real projective space. We will characterize which of those representations give convex-cocompacts subgroups of the real projective space. Convex cocompact subgroups of the real projective space should be thought of as generalizations of convex cocompact subgroups of the isometry group of the hyperbolic space. Joint work with Jeff Danciger, François Guéritaud, Fanny Kassel and Gye-Seon Lee. |
| Résumé: |
In recent years the unitary representation theory of automorphism groups of trees has garnered a relatively large amount of attention. The focus has largely been on the representation theory of non-amenable automorphism groups that satisfy certain vertex-transitivity assumptions. In this setting, the picture that is unfolding suggests that there is a strong correspondence between an automorphism group being type I (or CCR) and the action of the group on the boundary of the tree being highly transitive.
The non-compact amenable automorphism groups of trees have not received as much attention as their non-amenable counterparts, at least in regards to their representation theory. This talk will focus on discussing an ongoing project studying the unitary representations of certain classes of non-compact amenable groups acting on trees. In particular, we study automorphism groups of regular trees that are closed, fix an end of the tree, and are transitive on the vertices; such groups are called scale groups. A large class of scale groups arise by taking the semi-direct product of a closed totally disconnected locally compact contraction group with the integers. Thus, understanding the unitary representations of contraction groups and the application of the Mackey little group method in this setting is critical to understanding the representation theory of such scale groups. In this talk I will discuss new examples of amenable type I and non type I groups acting on trees arising from groups of the form described above. The talk will give a basic overview of scale groups and contraction groups, as well as unitary representation theory and the Mackey little group method. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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 $\Gamma$. |
| 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. |
| 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. |
| 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. |