| Résumé: |
To any reductive group $\mathbb G$ (such as $SL_n$) one can associate an affine flag variety $\mathcal X$, whose geometry is related to the representation theory of $\mathbb G$ and of its loop group $\mathbb G[t,t^{-1}]$. Kazhdan-Lusztig R-polynomials relate some of the structure of $\mathcal X$ to the combinatorics of a Coxeter group associated to $\mathbb G$, namely its affine Weyl group. These polynomials are a cornerstone in the famous affine Kazhdan-Lusztig theory. If we try to replace $\mathbb G$ by a Kac-Moody, non-reductive group, $\mathcal X$ can still be defined, but has no reasonable topology, and some of its structure is lost: there is an analog $W^+$ of the affine Weyl group, but which is only a semi-group, and has no proper Coxeter structure. However in 2016 Braverman Kazhdan and Patnaik have introduced a partial order on $W^+$ which could play the role of the Bruhat order. Since then, some key combinatorial properties of this semi-group have been obtained, making the definition of affine Kazhdan-Lusztig R-polynomials in this context reasonable.
In my talk, after introducing these polynomials in the reductive setting, I will present a path model construction for affine Kazhdan-Lusztig R polynomials associated to Kac-Moody groups. This path model was schemed in a prepublication of Muthiah in 2019, and relies on later work of Bardy-Panse, Hébert and Rousseau on twin masures, which we will use as a black box. This is a joint work with A. Hébert. |
| Résumé: | In this talk, I will discuss unitary representations of some non-locally-compact Polish groups acting on regular trees. Let $d$ be a cardinal either finite or countable and let $T$ be the regular tree of arity $d$. Given a permutation group $G$ on a set $\Omega$ of cardinality $d$, the associated universal Burger-Mozes group $U(G)$ is the largest subgroup of $\mathrm{Aut}(T)$ which, on the edges around every fixed vertex $T$, acts like $G$ does on $\Omega$. In finite arity, $U(G)$ is locally compact and its irreducible unitary representations were classified by Olshanskii, assuming that $G$ is 2-transitive. I will discuss how to extend Olshanskii’s classification to the infinite arity case, assuming that $G$ is 2-transitive and oligomorphic. |
| Résumé: | TBA |
| Résumé: | To any locally compact group $G$, one can associate a von Neumann algebra $L(G)$, generated by the left regular representation of $G$. This algebra reflects the decomposition properties of the representation: $L(G)$ is a factor — i.e., has trivial center — if and only if the regular representation does not split as a direct sum of two disjoint subrepresentations. In the discrete case, Murray and von Neumann showed that $L(G)$ is a factor if and only if all non-trivial conjugacy classes are infinite. By contrast, for non-discrete groups, determining factoriality becomes more subtle. In this talk, we address this question for totally disconnected groups. We present a sufficient criterion for factoriality, apply it to certain groups acting on (boundaries of) trees, and discuss the consequences of these results for the type of the group and its representations. |
| Résumé: | In recent work with Chesebro and Martin we classified Kleinian groups generated by two finite order elements. A subclass of these are the generalised triangle groups which have one additional relator. Jointly with current Masters student Csizmadia and former Masters student Carroll, using the machinery the latter developed in his PhD, we determine resolutions and cohomology groups for these. |
| Résumé: | In 1984 Cannon showed that hyperbolic groups have finitely many cone types. In this talk, I will demonstrate how this result can be extended to non-positively curved $k$-fold triangle groups. I will explain how this implies that such groups have an automatic structure and discuss some applications. |
| Résumé: | We propose to study actions of countable groups on measure spaces such that every group element individually acts as a conservative transformation, that is, a transformation such that no set of positive measure is disjoint from all its translates. We construct such actions of the free group, using the measurable full group of a hyperfinite equivalence relation. The motivation for our work was to find interesting examples of boomerang subgroups. Based on joint work with Y. Glasner and T. Hartnick. |
| Résumé: |
Generalized root systems (GRS) unite different generalizations of root systems in one concept. We call a finite, non-empty subset $\Phi$ of a Euclidean vector space $E$ a GRS if for every $\alpha,\beta\in\Phi$:
My talk will concern my current work on proving this same result in an elementary fashion without relying on advanced results. |
| Résumé: | Like standard growth of (finitely generated) groups, one can define conjugacy growth of groups which, informally, counts the number of conjugacy classes in a ball of radius $n$ in a Cayley graph. This was first studied by Riven for free groups, and techniques from geometry, combinatorics and formal language theory have proven to be useful for determining information about the conjugacy growth series for a variety of groups. In this talk we will survey these techniques and, in joint work with Laura Ciobanu, determine the conjugacy growth for dihedral Artin groups. |
| Résumé: | (Joint work with Y. Barnea, M. Ershov, A. Le Boudec, M. Vannacci and Th. Weigel.) The commensurator of a group is a group that encapsulates (up to a suitable equivalence) all isomorphisms between finite index subgroups of the group. We study the commensurator of a free group $F$ and of a free pro-$p$ group, and also the $p$-commensurator of $F$ (which is the subgroup of the commensurator that respects the pro-$p$ topology on $F$), with a focus on normal subgroup structure. As well as 'global' results about the commensurator as a whole, we obtain some new constructions of simple groups: finitely generated simple groups with a free commensurated subgroup, and nondiscrete compactly generated simple locally compact groups that possibly have a free pro-$p$ open subgroup. |
| Résumé: | We study the centralizer of a parabolic subalgebra in the Hecke algebra associated with an arbitrary (possibly infinite) Coxeter group. While the center and cocenter have been extensively studied in the finite and affine cases, much less is known in the indefinite setting. We describe a basis for the centralizer, generalizing known results about the center. Our approach combines algebraic techniques with geometric tools from the Davis complex, a CAT(0) space associated with the Coxeter group. As part of the construction, we classify finite partial conjugacy classes in infinite Coxeter groups and define a variant of the class polynomial adapted to the centralizer. |
| Résumé: |
Right-angled Artin groups (RAAGs) form a family of finitely generated groups that play an important role in geometric and computational group theory. Their study is often related to the one of their universal Salvetti complex, a CAT(0)-cube complex on which they act geometrically.
In this talk, we introduce a generalisation of RAAGs to topological groups based on the notion of generalised presentations. Remarkably, those groups have a cellular action on a thicker version of the universal Salvetti complex of a RAAG with controlled cell stabilisers. Due to the high connectivity of this complex, we can deduce homological/homotopical finiteness properties for topological RAAGs. On the way, we will also discuss the connection of this complex with Cayley-Abels graphs of the relevant groups, and describe a canonical building-like structure on it. Work in progress with I. Castellano, B. Nucinkis, and Y. Santos Rego. |
| Résumé: |
A group, $\mathcal{H}$, of automorphisms of a totally disconnected locally compact group $G$ for which there is a common tidy subgroup, $U$, is called flat. The subgroup of $\mathcal{H}$ stabilising $U$ is the uniscalar subgroup. A flat group of automorphisms is free abelian modulo its uniscalar subgroup.
This talk will outline an approach to proving this that is more direct than the one first given. The argument factors $U$ into subgroups, $U_\rho$, that are either expanded or contracted by elements of $\mathcal{H}$. Each factor consequently is dilated by $\mathcal{H}$ to a closed subgroup $\widetilde{U}_\rho\leq G$ and the logarithm of the modular function of $\mathcal{H}|_{\widetilde{U}_\rho}$ determines a homomorphism $\rho : \mathcal{H}\to\mathbf{Z}$. The homomorphisms $\rho$ and subgroups $U_\rho$ correspond to roots and root subgroups in the case where $G$ is a semisimple $p$-adic Lie group. In the general case, a version of commutation relations between root subgroups holds. |