| Résumé: | A trivalent triangle group is a triangle of groups with the trivial face group and edge groups isomorphic to $\mathbb Z/3\mathbb Z$. In this talk, I will introduce the weighted $L^2$-cohomology of Coxeter groups originally defined by Davis, Dymara, Januszkiewicz and Okun and use it to show virtual fibration and consequently incoherence of certain trivalent triangle groups acting on hyperbolic buildings. |
| Résumé: | TBA |
| Résumé: | Let $\Phi$ be a finite root system. A $\Phi$-graded group is a group $G$ together with a family of subgroups $(U_\alpha)_{\alpha \in \Phi}$ satisfying some purely combinatorial axioms. The main examples of such groups are the Chevalley groups of type $\Phi$, which are defined over commutative rings and which satisfy the well-known Chevalley commutator formula. We show that if $\Phi$ is of rank at least 3, then every $\Phi$-graded group is defined over some algebraic structure (e.g. a ring, possibly non-commutative or, in low ranks, even non-associative) such that a generalised version of the Chevalley commutator formula is satisfied. A new computational method called the blueprint technique is crucial in overcoming certain problems in characteristic 2. This method is inspired by a paper of Ronan-Tits. |
| Résumé: | Given a cocompact CAT(0) cube complex, we study the group of its cubical isometries, which frequently forms a non-discrete tdlc group. We present a method to study these groups that is focused on our ability to understand the stabilizer subgroups. We demonstrate the potency of this method by introducing a finite, topologically generating set and discuss an important simple subgroup. If there is time, we discuss some open questions regarding the placement of these groups among non-discrete tdlc groups. |
| Résumé: | The rank of a finitely generated group is the minimal size of a generating set. Several questions that received a lot of attention around 50 years ago ask about the rank of finitely generated groups, and how this relates to the rank of their direct powers. In this context, Wiegold asked about the existence of infinite simple characteristic quotients of free groups. I will review this framework, ask several questions (old and new) and present a solution to Wiegold’s problem, joint with Rémi Coulon. |
| Résumé: | We will review some general facts about isometric representations of groups in the infinite-dimensional hyperbolic spaces and we will introduce two families of functions which play a role with respect to hyperbolic representations analogous to that played by the functions of positive type with respect to unitary representations. Using these families we will construct a continuum of non-equivalent hyperbolic representations for any non-elementary subgroup of $\mathrm{PO}(1,n)$ and $\mathrm{PU}(1,n)$ and we will describe the implication of this fact towards classification questions. |
| Résumé: | Scale theory, largely developed by George Willis and his collaborators, plays a major role in the theory of totally disconnected, locally compact groups. It is a powerful set of tools for analysing the dynamics of conjugation by a single element, or more generally by a flat subgroup, where 'flat' can be thought of as meaning 'approximately abelian'. Scale theory methods are quite general but also abstract, so it would be useful to find geometric interpretations of scale theory concepts when the group acts (in a nice way) on a (nice) metric space. In particular, since scale theory is vacuous for discrete groups, this ties in with geometric properties of actions that can only occur when the group is nondiscrete. I will present some work in progress, principally on the case where G acts properly and semisimply on a complete CAT(0) space. |
| Résumé: | It is a classical question in abstract harmonic analysis to determine which locally compact groups are type I. Many well known classes of groups satisfy the type I property such as semisimple Lie groups and reductive algebraic groups over local fields. Recent research in the unitary representation theory of totally disconnected locally compact (tdlc) groups is interested in determining which automorphism groups of regular trees are type I. The focus so far has largely been on non-amenable automorphism groups of trees. I will talk about recent work that studies the unitary representation theory of scale groups. Scale groups are a class of amenable groups that arise from any non-uniscalar tdlc group and they naturally act on a regular tree as a fixator of a boundary point. I provide (non-)type I results for scale groups that are further assumed to be contractive. The results naturally contrast the results from the non-amenable setting. The arguments require heavy use of the structure theory of locally compact contraction groups and Mackey’s little subgroup analysis. I will give an introduction to all of these topics during the talk. |
| Résumé: | We say that a group has Kazhdan’s property (T) if the trivial representation is isolated in its unitary dual. If a given group has this property, it is interesting to search for representations which are, in some sense, the closest to the trivial representation. In this talk, I will explain how to identify such a representation among boundary representations of groups acting on $\widetilde{A}_2$ buildings. |
| Résumé: |
Say we are given only the $R$-algebra structure of a group ring $RG$ of a finite group $G$ over a commutative ring $R$. Can we then find the isomorphism type of $G$ as a group? This so-called Isomorphism Problem has obvious negative answers, considering e.g. abelian groups over the complex numbers, but more specific formulations have led to many deep results and beautiful mathematics. The last classical open formulation was the so-called Modular Isomorphism Problem: Does the isomorphism type of $kG$ as a ring determine the isomorphism type of $G$ as a group, if $G$ is a $p$-group and $k$ a field of characteristic $p$?
Starting with an overview on the state of knowledge on general Isomorphism Problems and the modular one in particular, I will present a negative solution found in 2021 with D. García-Lucas and Á. del Río as well as a generalization obtained recently with T. Sakurai, but also positive structural results and several problems remaining open. |
| Résumé: | Let $\mathfrak{l}$ be a real form of a complex semisimple Lie algebra. If $\mathfrak{l}$ is of compact type, the usual (Drinfeld-Jimbo) quantum universal enveloping algebra $\mathcal{U}_q(\mathfrak{l})$ of $\mathfrak{l}$ has an associated quantum group $C^{\ast}$-algebra $C^{\ast}_q(L)$, and there is a nice interplay between the representation theories of these two objects. By contrast, when $\mathfrak{l}$ is non-compact, a path to such an associated object is so far unclear, especially from the representation theoretical viewpoint. In the first part of the talk, we sketch the general setting in the compact case. Subsequently, we use the example of $\mathfrak{sl}(2,\mathbb{R})$ to suggest a different way of quantising the universal enveloping algebra $\mathcal{U}(\mathfrak{sl}(2,\mathbb{R}))$, making the integration problem more approachable. We conclude by comparing the representations of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ to their classical analogues. |