Totally Disconnected Locally Compact Groups

from a Geometric Perspective

from a Geometric Perspective

Abstracts

Pierre-Emmanuel Caprace, UC Louvain

Simple p-adic Lie groups with abelian Lie algebras

This talk aims at describing the construction of second countable,
topologically simple p-adic Lie groups of arbitrary dimension with
abelian Lie algebras. This is relevant to the general structure theory
of

*p*-adic Lie groups will be explained. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups. Based on joint work with Ashot Minasyan and Denis Osin.
Ilaria Castellano, Bielefeld University

Coxeter groups and t.d.l.c. groups acting on buildings

For totally disconnected groups which are closed automorphism groups of locally finite
buildings with sufficiently transitive action, we investigate two invariants, the rational discrete cohomological dimension and the number of ends. We show that these invariants are related to those of the Weyl groups.
Joint work with Bianca Marchionna and
Thomas Weigel, Università di Milano Bicocca.

Ged Corob Cook, University of Lincoln

Finiteness conditions for actions of totally disconnected, locally compact groups

One natural approach to studying totally disconnected, locally compact (tdlc) groups is by investigating their actions on CW-complexes with compact open cell stabilisers. Arising naturally as higher analogues of compact generation and presentation for these groups, this investigation suggests ‘type

*F*’ and ‘type_{n}*FP*’ conditions, similarly to these conditions for abstract groups. An obstacle is that the category of discrete modules for tdlc groups over_{n}**Z**does not have enough projectives. There are two possible remedies for this: working over**Q**, and working with categories of topological modules. These two ideas allow us to reprove, in the tdlc case, a lot of the usual results on finiteness conditions for abstract groups.
Yves de Cornulier, CNRS – University of Nantes

Locally compact modules over abelian groups

Let

*A*be a compactly genenerated abelian group. We present structural results on locally compact*A*-modules (= locally compact abelian groups*M*endowed with a continuous action of*A*). Classical commutative algebra addresses the case when both*A*and*M*are discrete. On difficulty in the non-discrete case is the lack of Noetherianity. However we obtain a weak form thereof: if*M*is compactly generated and (*M*) is an ascending sequence of submodules, and_{n}*N*the closure of its union, the*N/M*is compact for_{n}*n*large enough. In particular, every closed submodule of a compactly generated module is compactly generated. The structural result make use of the now classical Willis theory. As an application, one can extend to compactly generated LC metabelian groups known results about discrete ones, such as: the existence of a maximal compact normal subgroup, the Bieri-Strebel criterion for compact presentability, etc.
Tom De Medts, Ghent University

TDLC groups acting on right-angled buildings

Groups acting on trees provide a rich source of topological groups. Our starting point are the (by now well-known) universal groups with a prescribed local action, as introduced by Marc Burger and Shahar Mozes. Following a suggestion of Pierre-Emmanuel Caprace, we have extended this idea to the setting of right-angled buildings, a richer family of geometric objects. We will give an overview of the results, focusing on the interplay between local and global structure, in which the combinatorics of the underlying right-angled Coxeter group often plays an unexpected role.
This is based on joint work with Jens Bossaert, Ana C. Silva and Koen Struyve.

Michal Ferov, University of Newcastle

Automorphism groups of Cayley graphs of Coxeter groups

It is well known that groups of automorphisms of connected vertex-transitive locally-finite graphs are compactly generated totally disconnected locally compact groups. However, given such graph, it hard to decide whether its automorphism group is discrete or not. We use the machinery of local actions to characterise Coxeter groups whose Cayley graph, with respect to the standard generating set, is has non-discrete group of automorphisms. As an application, we construct examples of infinitely generated Coxeter groups whose Cayley graphs have unusual properties.
Joint work with Federico Berlai.

Alejandra Garrido, Universidad Complutense de Madrid

Simple tdlc full groups

Full groups of homeomorphisms of the Cantor space are a rich source of simple groups.
Given the right conditions on the "seed" group, the full group can be turned into a tdlc group that contains a simple compactly generated subgroup. An example of this are tree almost automorphism groups.
This construction and the conditions under which it can be fruitfully used has general theoretical consequences for the class of topologically simple, non-discrete, compactly generated, tdlc groups.

Helge Glöckner, Paderborn University

Totally disconnected contraction groups

An automorphism alpha of a topological group

*G*is called contractive if all alpha-orbits converge to the neutral element. A pair (*G*,α) of a locally compact group and a contractive automorphism is called a locally compact contraction group. As shown by Siebert, every locally compact contraction group is a direct product of a simply connected nilpotent Lie group and a totally disconnected, locally compact contraction group. In the talk, I shall describe results concerning the latter groups, obtained in joint work with George A. Willis. The most recent results concern the structure of contraction groups which are torsion groups.
Adrien Le Boudec, ENS Lyon

Commability and graphs of abelian groups

Two locally compact groups

*H,G*such that there exists a continuous morphism from*H*to*G*with compact kernel and with closed cocompact image are commable. Among locally compact groups, commability is the equivalence relation generated by this property. For instance two discrete groups that are cocompact lattices in the same locally compact group are commable. In general, determining the commability class of a certain discrete group Γ is a challenging task. It is an extension of the problem of determining its cocompact envelopes (those locally compact groups containing Γ as a cocompact lattice). In this talk, we address this problem for certain groups acting on trees. We consider the class of groups admitting a cocompact action on a locally finite tree, with vertex stabilizers virtually free abelian of rank*n*≥1. In the case*n*=1, Baumslag-Solitar groups are examples of such groups. The behavior of that class of groups with respect to quasi-isometries is described by works of Mosher-Sageev-Whyte, Farb-Mosher, and Whyte. We prove that many groups in this class are not commable (although they are quasi-isometric by Whyte's result). Joint work with Yves Cornulier.
Waltraud Lederle, UC Louvain

Compact IRS and URS

Let

*G*be a locally compact group. We prove that every URS consisting of compact subgroups is contained in a compact, normal subgroup. A similar question for IRS is still very much work in progress. I will explain what we already know and where Cayley-Abels graph come into play. No prior knowledge about IRS or URS is required. Based on joint work with Pierre-Emmanuel Caprace, Tal Cohen, Gil Goffer and Todor Tsankov.
Rögnvaldur Möller, University of Iceland

Minimal degree of a Cayley-Abels graph of a totally disconnected locally compact group

A compactly generated, totally disconnected, locally compact group can be made to act transitively on a connected, locally finite graph such that stabilizers of vertices are compact, open subgroups. Such a graph is called a Cayley-Abels graph for the group. In this talk I want to describe how the minimal degree of a Cayley-Abels graph for a given group relates to other properties if the group. I also want to discuss what can be said about the graphs that realize the minimal degree and how the group acts on them.
Joint work with Arnbjörg Soffía Árnadóttir (Waterloo (and now Copenhagen)) and Waltraud Lederle (Louvain-la-Neuve).

Miho Mukohara, University of Tokyo

C*-simplicity of tdlc groups

A locally compact group

*G*is called C*-simple if its reduced group C*-algebra is simple. C*-simplicity of discrete groups is well studied and sufficiently characterisations are known. On the other hand, the study of non-discrete C*-simple groups is still in the development stage and there are few examples. It is known that every C*-simple group is totally disconnected. In this talk I’ll introduce a sufficient condition for C*-simplicity and non-discrete examples of C*-simple groups constructed by HNN extensions and amalgamated free products.
Tatiana Nagnibeda, University of Geneva

Schreier graphs of self-similar actions: applications to spectral graph theory

Schreier graphs of self-similar actions have proved useful in the study of the groups that define them but are also interesting in their own right, in particular they serve as a rich class of new examples in spectral graph theory studying spectra and spectral measures associated with random walks.

Yury Neretin, University of Vienna

Polyhomomorphisms of locally compact groups

Let

*G*,*H*be unimodular locally compact groups with fixed normalizations of Haar measures γ and η. A polyhomomorphism*G*→*H*is a closed subgroup*R*⊂*G*×*H*with fixed normalization of the Haar measure such that pushforwards of measure ρ to*G*and*H*are dominated by γ and η respectively. The set of all polyhomomorphisms*G*→*H*is a compact space with respect to the Chabauty topology. For polyhomomorphisms*G*→_{1}*G*and_{2}*G*→_{2}*G*there is a well-defined product_{3}*G*→_{1}*G*, so we get a category of polyhomomorphisms. We discuss some properties of this category and some examples._{3}
Colin Reid, University of Newcastle

Growing trees from compact subgroups

Given a noncompact compactly generated totally disconnected locally compact (t.d.l.c.) group

*G*, I will give a sufficient condition in terms of compact subgroups for*G*to act on a tree with compact open arc stabilizers and with nontrivial rigid stabilizers of half-trees. In particular, it follows that*G*has infinitely many ends. I will sketch an application to the local structure of Kac–Moody groups, specifically the question of whether any open subgroup splits as a direct product. Joint work with Pierre-Emmanuel Caprace and Timothée Marquis.
Rachel Skipper, University of Utah

Computing scale in Neretin's group

For an automorphism of a totally disconnected, locally compact group, Willis introduced the notion of scale which provides information on its behavior. In this talk, we will discuss the setting where the tdlc group is Neretin's group and where the automorphism comes from conjugation in the group. This is an ongoing joint work with Michal Ferov and George Willis at the University of Newcastle.

Simon Smith, University of Lincoln

Local action diagrams and Tits' Independence Property (P)

In this talk, which concerns groups acting on trees, I will describe joint work with Colin Reid in which we develop a `local action' complement to classical Bass--Serre theory. We call this the theory of local action diagrams. To highlight the significance of our theory, I will describe how we can use it to obtain a complete description and classification of all closed group actions on trees with Tits' Independence Property (P).

Katrin Tent, University of Münster

Simplicity of automorphism groups of homogeneous structures

Automorphism groups of homogeneous structures are totally disconnected, albeit often not locally compact. We discuss some general criteria that can be used to show that the automorphism group of a homogeneous structure (such as metric space, right-angled building, graph or hypergraphs) are simple groups or have simple quotients.

Federico Vigolo, Göttingen University

On coarse geometry and group theory

Coarse geometry is the paradigm of studying the geometric features of a space which are stable under bounded perturbations. Attempting to construct coarse geometric structures on groups naturally leads to the notion of coarse groups (these are group objects in the category of coarse spaces). In this talk I will explain this thought process, illustrating some of the difficulties and problems that one encounters along the way. The overarching goal is to explore some phenomena, questions and connections between the theory of coarse groups and classical (topological) group theory.

George Willis, University of Newcastle

Scale groups

A scale group is a closed subgroup of the isometry group of a locally finite tree which fixes an end of the tree and is transitive on the vertices. If

*G*is a t.d.l.c. group and*x*∈*G*has scale,*s*(*x*), greater than 1, then a certain closed subgroup of*G*containing*x*acts as a scale group on the regular tree with valency*s*(*x*)+1. Scale groups are thus concretely described groups which appear as subquotients of abstract t.d.l.c. groups. There is also a close correspondence between scale groups and self-replicating groups acting on rooted trees. The connection between scale groups, the scale and self-replicating groups will be explained in the talk, followed by questions about the class of scale groups.