Tag - Locally compact groups

Emma Brink: Condensed Group Cohomology

18th April, 2023

The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of G, M (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of π0(G) with coefficients in M.

Generalizing results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological δ-functor in the condensed setting for a large class of topological groups.

Matthew Conder: Discrete two-generator subgroups of PSL2(ℚp)

1st July, 2022

Due to work of Gilman, Rosenberger, Purzitsky and many others, discrete two-generator subgroups of PSL2(ℝ) have been completely classified by studying their action by Möbius transformations on the hyperbolic plane. Here we aim to classify discrete two-generator subgroups of PSL2(ℚp) by studying their action by isometries on the Bruhat-Tits tree. We first give a general structure theorem for two-generator groups acting by isometries on a tree, which relies on certain Klein-Maskit combination theorems. We will then discuss how this theorem can be applied to determine discreteness of a two-generator subgroup of PSL2(ℚp).

Max Carter: Unitary representations and the type I property of groups acting on trees

1st April, 2022

Unitary representations are a classical and useful tool for studying locally compact groups: motivated in part by quantum mechanics, they have been studied in detail since the early-mid 1900s with much success, and they enable group theorists to employ functional analytic techniques in the study of locally compact groups. The algebras that unitary representations generate play an important role in not only understanding the representation theory of a locally compact group, but also in understanding properties pertaining to the group itself. This talk will give a brief introduction to some of the basics of the unitary representation theory of locally compact groups, with focus placed on the associated operator algebraic structures/properties. In particular, 'type I groups' and 'CCR groups' will be the main focus. As an application, I will discuss some current research interests in the unitary representation theory of groups acting on trees, including work of myself on the unitary representation theory of 'scale groups'.

Michal Ferov: Automorphism groups of Cayley graphs of Coxeter groups: when are they discrete?

4th March, 2022

The group of automorphisms of a connected locally finite graph is naturally a totally disconnected locally compact topological group, when equipped with the permutation topology. It therefore makes sense to ask for which graphs is the topology not discrete. We show that in case of Cayley graphs of Coxeter groups, one can fully characterize the discrete ones in terms of the symmetries of the corresponding Coxeter system.

Colin Reid: Groups acting on trees with prescribed local actions

23rd September, 2021

Actions on trees are ubiquitous in group theory. The standard approach to describing them is known as Bass–Serre theory, which presents the group acting on the tree as assembled from its vertex and edge stabilizers. However, a different approach emerges if instead of considering vertex and edge stabilizers as a whole, we focus on local actions, that is, the action of a vertex stabilizer only on the immediate neighbours of that vertex. Groups acting on trees defined by their local actions are especially important as a source of examples of simple totally disconnected locally compact groups, with a history going back to a 1970 paper of Tits. I will go through some highlights of this theory and then present some recent joint work with Simon Smith: we develop a counterpart to Bass–Serre theory for local actions, which describes all possible local action structures of group actions on trees.

George Willis: Constructing groups with flat-rank greater than 1

30th August, 2021

The contraction subgroup for x in the locally compact group, G,

con(x)={ g ∈ G ∣ xngx−n → 1 as n → ∞ },

and the Levi subgroup is

lev(x)={ g ∈ G ∣ {xngx−n}n∈ℤ has compact closure }.

The following will be shown. Let G be a totally disconnected, locally compact group and x ∈ G. Let y ∈ lev(x). Then there are x′ ∈ G and a compact subgroup, K ≤ G such that:
-K is normalized by x′ and y,
-con(x′)=con(x) and lev(x′)=lev(x) and
-the group ⟨x′,y,K⟩ is abelian modulo K, and hence flat.
If no compact open subgroup of G normalized by x and no compact open subgroup of lev(x) normalized by y, then the flat-rank of ⟨x′,y,K⟩ is equal to 2.

Sven Raum: Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

9th August, 2021

In the 90s, Nebbia conjectured that a group of tree automorphisms acting transitively on the tree's boundary must be of type I, that is, its unitary representations can in principal be classified. For key examples, such as Burger-Mozes groups, this conjecture is verified. Aiming for a better understanding of Nebbia's conjecture and a better understanding of representation theory of groups acting on trees, it is natural to ask whether there is a characterization of type I groups acting on trees. In 2016, we introduced in collaboration with Cyril Houdayer a refinement of Nebbia's conjecture to a trichotomy, opposing type I groups with groups whose von Neumann algebra is non-amenable. For large classes of groups, including Burger-Mozes groups, we could verify this trichotomy.

In this talk, I will motivate and introduce the conjecture trichotomy for groups acting on tress and explain how von Neumann algebraic techniques enter the picture.

Lancelot Semal: Unitary representations of totally disconnected locally compact groups satisfying Ol’shanskii’s factorization

5th June, 2021

We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings.

Colin Reid: Abelian chief factors of locally compact groups

9th December, 2020

Recent work in the theory of locally compact second-countable (l.c.s.c.) groups has highlighted the importance of chief factors, meaning pairs of closed normal subgroups K/L such that no closed normal subgroups lie strictly between K and L. In particular, the group K/L is then topologically characteristically simple, meaning it has no proper nontrivial closed subgroup that is preserved by all automorphisms. I will present a classification of the abelian l.c.s.c. topologically characteristically simple groups: these all occur as chief factors of soluble groups, and naturally fall into five families with a few parameters. Each family has a straightforward characterization within the class of abelian l.c.s.c. groups, without directly invoking the property of being topologically characteristically simple.

Stephan Tornier: Think globally, act locally

9th December, 2020

Let G be a group acting on a regular tree. The 'local' actions that vertex stabilisers in G induce on balls around the fixed vertex are innately connected to the 'global' structure of G. I demonstrate this relationship and define a particularly accessible class of groups acting on (locally finite) regular trees by 'prescribing' said local actions, following Burger-Mozes. Being defined solely in terms of finite permutation groups, these groups allow us to introduce computational methods to the world of locally compact groups: I will outline the capabilities of a recently developed GAP package that provides methods to create, analyse and find suitable local actions.