Totally Disconnected Locally Compact Groups:

Local to Global

Local to Global

Abstracts

Sebastian Bischof, University of Gießen

Groups of Kac-Moody type over

*F*_{2}
In this talk we will work out necessary and sufficient conditions for the existence of groups of Kac-Moody type over

*F*with prescribed commutator relations. In particular, we give a construction of groups of Kac-Moody type over_{2}*F*of type (4,4,4), whose existence answers a question of Tits from the 1990s. In the end we will discuss several applications. Using a result of Caprace and Rémy, we obtain uncountable many new examples of topologically simple, compactly generated tdlc groups._{2}
Michal Doucha, Czech Academy of Sciences

Groups acting generically on the Cantor space

Following Hochman, say that a locally compact second countable (and without loss of generality, totally disconnected) group

*G*has the Strong topological Rokhlin property (STRP) if*G*admits a generic action on the Cantor space, i.e. a continuous action whose conjugacy class is comeager in the Polish space of all continuous actions of*G*on the Cantor space. The aim of my talk is to give a characterization of such groups in symbolic dynamical terms based on certain special class of sofic subshifts. I will first focus on discrete groups for which the characterization is complete and then move on to the general case. The latter is work in progress, however I will provide a characterization of locally compact pro-countable Polish groups that have the STRP.
Benjamin Klopsch, University of Düsseldorf

Hausdorff dimension in finitely generated pro-p groups - old and new

Every finitely generated pro-p group

*G*comes equipped with a range of translation-invariant metrics that are naturally induced by filtration series such as the*p*-power filtration or the lower*p*-series. Given such a metric, the distribution of closed subgroups in*G*gives rise to a corresponding Hausdorff spectrum. It is a long-standing open question whether the finiteness of the Hausdorff spectrum, with respect to the*p*-power filtration, say, implies that the pro-*p*group*G*is*p*-adic analytic. In my talk I will give a brief introduction to the subject, report on some of the progress made during the past five years and present recent joint work with A. Thillaisundaram and I. de las Heras about the Hausdorff spectra of*p*-adic analytic groups with respect to the lower*p*-series.
Waltraud Lederle, UC Louvain

Boomerang subgroups

The set of subgroups of a locally compact group, with an appropriate topology, is called its Chabauty space. The group acts on it by conjugation. We call a subgroup a boomerang subgroup if, considered as element in the Chabauty space, it is strongly recurrent in a specific sense. Poincaré recurrence then implies that every IRS gives full measure to the set of boomerang subgroups. Therefore, if a statement is almost surely true for every IRS, we can hope that it is deterministically true for every boomerang subgroup. We present a few results in this direction.
This is joint work with Yair Glasner.

Rögnvaldur Möller, University of Iceland

Graphs of group actions

Joint work with Florian Lehner (University of Auckland), Christian Lindorfer (TU Graz) and Wolfgang Woess (TU Graz).
This is a preliminary report on work in progress.
The concept of a graph of group actions and its fundamental group is inspired by Bass-Serre theory and the Burger-Mozes construction. Using this construction we recover the fundamental group of a graph of groups in Bass-Serre theory, the Burger-Mozes groups and some more in addition. Using this approach it is possible to construct groups with various conditions on the local action without the group neccessarily having Property (P).

Volodymyr Nekrashevych, Texas A&M University

Groups from groupoids

Groupoids are well suited for the study of local symmetries of various structures. We will give a short overview of the main notions related to topological groupoids, and then discuss various groups constructed using them. We will show how topological properties of groupoids can be used to prove large-scale properties of the associated groups (such as simplicity, amenability, growth, etc.).

André Nies, University of Auckland

Representing totally disconnected, locally compact groups by countable structures

All topological groups will be countably based. A totally disconnected, locally compact (tdlc) group has only countably many compact open cosets. We consider the structure with domain the collection of such cosets, as well as the empty set. It is equipped with the intersection operation and a natural groupoid operation given by the product of appropriate cosets.
We obtain a Borel duality between the tdlc groups and such “meet groupoids”. This can be used, for instance, to understand the topological automorphism group of a tdlc group, as well as its Chabauty space. We mainly use this duality to study computability of a tdlc group. We show that given a computable presentation of a t.d.l.c. group, the modular function and the Cayley-Abels graphs (in the compactly generated case) are computable. We give an example (joint with G. Willis) where the scale function fails to be computable.
The notion of a meet groupoid evolved from the coarse groups introduced by Kechris, N. and Tent, and further developed in the context of oligomorphic groups by N., Schlicht and Tent. The present work is joint with A. Melnikov.
References:

A. Kechris, A. Nies, and K. Tent. The complexity of topological group isomorphism. The Journal of Symbolic Logic 83.3 (2018): 1190-1203.

A. Melnikov and A. Nies. Computably totally disconnected locally compact groups. Preprint, 2022, https://arxiv.org/pdf/2204.09878.pdf

A. Nies, P. Schlicht, and K. Tent. Coarse groups, and the isomorphism problem for oligomorphic groups. Journal of Mathematical Logic 22.01 (2022): 2150029.

A. Kechris, A. Nies, and K. Tent. The complexity of topological group isomorphism. The Journal of Symbolic Logic 83.3 (2018): 1190-1203.

A. Melnikov and A. Nies. Computably totally disconnected locally compact groups. Preprint, 2022, https://arxiv.org/pdf/2204.09878.pdf

A. Nies, P. Schlicht, and K. Tent. Coarse groups, and the isomorphism problem for oligomorphic groups. Journal of Mathematical Logic 22.01 (2022): 2150029.

Piotr Nowak, IMPAN

Coboundary expanders and Gromov hyperbolicity

I will discuss the concept of higher-dimensional expanders as a generalization of expander graphs.
I will show that for a tower of residual finite coverings of a compact manifold, coboundary expansion in a certain dimension forces the fundamental group to be hyperbolic.
This is joint work with Dawid Kielak.

Sven Raum, University of Potsdam

On the unitary representation theory of groups acting on buildings of indefinite type

Unitary representation theory describes how groups act on complex Hilbert spaces. Motivated by physics and number theory, its focal point for about a century were Lie groups and algebraic groups. Their representation theory, while very complicated in practice, is theoretically rather well-behaved, meaning that classification results for irreducible representations are achievable. The continuously increasing interest in totally disconnected groups entailed the investigation of groups whose representation theory is "wild" in classical terms and which require new ideas in order to achieve satisfactory classification results. In this talk I will introduce the audience to this direction of research and survey results on the unitary representation theory of strongly transitive automorphism groups of buildings of indefinite type.

Colin Reid, University of Newcastle

Boundary-2-transitive actions on trees

Let

*T*be a locally finite tree. A natural class of 'large' closed subgroups*G*of Aut(*T*) are those with unbounded orbits that act transitively on the boundary (space of ends) of the tree. In fact, all such groups act 2-transitively on the boundary. I will talk about some reasons to be interested in this class of groups, and some restrictions I obtained on their structure in terms of local actions, in other words, the finite permutation groups induced by a subgroup fixing a vertex on the neighbours of that vertex. If one of the local actions of*G*has insoluble point stabilizers, then*G*has no prosoluble open subgroup and is micro-supported (with an exception that only occurs for the (31,21)-semiregular tree). If*G*is vertex-transitive, the local action of an end stabilizer is a point stabilizer of the local action of*G*; this is usually true also if*G*is not vertex-transitive, but there are some combinations of local actions where a local action of an end stabilizer can be smaller.
Yuhei Suzuki, Hokkaido Unievrsity

C*-simplicity of discrete extensions

After reviewing examples of non-discrete C*-simple groups, I will explain the following permanence property of the C*-simplicity:
C*-simplicity is preserved by extensions by discrete C*-simple groups.
Similar results hold true for the unique trace property.
For a proof, it is important to extend a result of Breuillard-Kalantar-Kennedy-Ozawa to the non-unital case.
It would be interesting that our analysis on tracial weights involves von Neumann algebra theory.
These generalizations also lead to a simpler and more general proof of Bryder-Kennedy's results on the reduced twisted crossed products.
Based on my preprint arXiv:2109.08606.

Stephan Tornier, University of Newcastle

A characterisation of discrete (

*P*)-closed groups acting on trees
Groups acting on trees play an important role in the general theory of locally compact groups for both theoretical and practical reasons. Attempts have been made to classify groups acting on trees that satisfy various transitivity conditions. For example, every closed, vertex-transitive subgroup of the automorphism group of a tree can be written as the intersection of a descending sequence of vertex-transitive (

*P*)-closed groups (_{k}*k*∈**N**). Here, being (*P*)-closed is a generalisation of Tits' original independence property (_{k}*P*)=(*P*). In recent work, Colin Reid and Simon Smith achieved an elegant parametrisation of general (_{1}*P*)-closed groups in terms of decorated graphs known as local action diagrams. We introduce this parametrisation and characterise those local action diagrams that result in discrete groups. (Joint work with Marcus Chijoff.)
Todor Tsankov, University of Lyon 1

Gleason complete flows of locally compact groups

The notion of an irreducible extension of a flow generalizes the one of an almost one-to-one extension (injective on a dense

*G*) and coincides with the one of a highly proximal extension for minimal flows. The existence of maximal such extensions was proved by Auslander and Glasner in the 70s for minimal flows using an abstract argument, and a concrete construction using near-ultrafilters was recently given by Zucker for arbitrary flows. When the acting group is discrete, the universal irreducible extension is nothing but the Stone space of the Boolean algebra of the regular open sets of the space, already considered by Gleason. We give yet another construction of the universal irreducible extension for arbitrary topological groups and prove that for such extensions (which we call Gleason complete) of a flow of a locally compact group_{δ}*G*, the stabilizer map*x→G*is continuous (for general flows, this map is only semi-continuous). This is a common generalization of a theorem of Frolík that the set of fixed points of a homeomorphism of a compact, extremally disconnected space is open and a theorem of Veech that the action of a locally compact group on its greatest ambit is free. This is joint work with Adrien Le Boudec._{x}
Stefaan Vaes, KU Leuven

Measure equivalence embeddings of free groups and free group factors

I will present a simple and explicit proof that the free group

*F*admits a measure equivalence embedding into any nonamenable locally compact second countable group. I will also explain a new II_{2}_{1}factor analogue for the concept of measure equivalence embeddings and discuss its basic properties. In particular, we prove that a II_{1}factor*M*is nonamenable if and only if the free group factor*L*(*F*) admits such a measure equivalence embedding into_{2}*M*. The talk is based on recent collaborations with Tey Berendschot and Daniel Drimbe.
Alain Valette, University of Neuchâtel

Limits of Cartan subgroups in the Chabauty space of SL

_{n}(**Q**_{p})
After a general introduction (with examples) to the Chabauty space of closed subgroups in a given locally compact group, we will focus on the following question: let

*C*be the Cartan subgroup of diagonal matrices in SL_{n}(**Q**_{p}), can one describe the closure of the conjugacy class of*C*in the Chabauty space of SL_{n}(**Q**_{p})? In joint work with C. Ciobotaru and A. Leitner, we show in particular that the closure consists of finitely many orbits for*n*≤ 4, but infinitely many for*n*≥ 7.
Matteo Vannucci, University of the Basque Country

Random subgroups in direct products of hereditarily just infinite groups

We study the probability that Haar-random elements generate an open subgroup in the direct product of finitely many hereditarily just infinite profinite groups. In particular, we show that two random elements in the direct product of finitely many absolutely simple simply connected split algebraic groups over local fields generate an open subgroup with probability one. This is joint work with Benjamin Klopsch and Davide Veronelli.

Thomas Weigel, Università di Milano-Bicocca

On some Dirichlet series associated to totally disconnected, locally compact groups

[abstract]

George Willis, University of Newcastle

Contraction groups and the scale

Contraction groups for elements of locally compact groups are important for understanding the structure and representations of these groups, while the scale is an integer-valued, positive continuous function which is a structural invariant for totally disconnected, locally compact groups. Both of these ideas concern the conjugation action of the group

*G*on neighbourhoods of the identity. They are directly linked by the fact that, for*g*∈*G*, the contraction group con(*g*) has non-compact closure if and only if*s*(*g*^{-1})>1. This link will be explained in the first part of the talk. The second part will describe the structure of contraction groups when they happen to be closed, including recent work with Helge Glöckner showing that locally pro-*p*contraction groups are nilpotent.