Schedule for: 21w5151 - Totally Disconnected Locally Compact Groups via Group Actions (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Let $A$ and $B$ be two permutations in $Sym(n)$ which "almost commute"- are they a small deformation of permutations that truly commute? More generally, if $R$ is a system of wards-equations in variables $X=x_1,\dots,x_d$ and $A_1,\dots,A_d$ permutations which are nearly solution; are they near true solutions? It turns out that the answer to this question depends only on the group presented by the generators $X$ and relations $R$. This leads to the notions of "stable groups" and "testable groups". We will present a few results and methods which were developed in recent years to check whether a group is stable\testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic and with IRS's ( =invariant random subgroups).

Let $G$ be a permutation group acting on a finite set $\Omega$. A subset $\mathcal{B}$ of $\Omega$ is called a base for $G$ if the pointwise stabilizer of $\mathcal{B}$ in $G$ is trivial. In the 19th century, bounding the order of a finite primitive permutation group $G$ was a problem that attracted a lot of attention. Early investigations of bases then arose because such a problem reduces to that of bounding the minimal size of a base of $G$. Some other far-reaching applications across Pure Mathematics led the study of the base size to be a crucial area of current research in permutation groups. In this part of the talk, we will investigate some of these applications and review some results about base size. We will present a recent improvement of a famous estimation due to Liebeck that estimates the base size of a primitive permutation group in terms of its degree. In the second part of the talk, we will define the concept of irredundant bases of $G$ and the concept of IBIS groups. Whereas bases of minimal size have been well studied, irredundant bases and IBIS groups have not yet received a similar degree of attention. Indeed, Cameron and Fon-Der-Flaas, already in 1995, defined such groups and proposed to classify some meaningful families. But only this year, a systematic investigation of primitive permutation IBIS groups has been started. We will discuss how we reduced the classification of primitive IBIS groups to the almost simple groups and affine groups. Eventually, we will conclude by mentioning recent advances towards a complete classification.

A permutation group $G$ on a set $X$ is called binary if the following condition holds: if $r > 2$ and $x,y\in X^r$ are $2$-equivalent $r$-tuples, then $x$ and $y$ must be in the same $G$-orbit. Here we say $x = (x_1,\dots,x_r)$ and $y = (y_1,\cdots,y_r)$ are $2$-equivalent if any pair $(x_i,x_j)$ can be mapped to the corresponding pair $(y_i,y_j)$ by an element of $G$. The definition was coined by Gregory Cherlin as part of his theory of homogeneous structures in model theory. Over 20 years ago, Cherlin conjectured that the all the finite primitive binary groups fall into three families: the full symmetric groups $Sym(X)$; cyclic groups of prime order; and a certain class of affine groups of dimension $1$ or $2$. In joint work with Nick Gill and Pablo Spiga, we have completed the proof of this conjecture. In the talk I will try to explain the point of the binary definition in relation to model theory, discuss various examples of binary groups, and indicate some of the strategies of the proof of the conjecture.

A classical result of Baer states that an element $x$ of a finite group $G$ is contained in the Fitting subgroup $F(G)$ of $G$ if and only if $x$ is a left Engel element of $G$. That is, $x \in F (G)$ if and only if there exists a positive integer $k$ such that $[g,_k x] := [g, x, . . . , x]$ (with $x$ appearing k times, and using the convention $[x_1 , x_2 , x_3 . . . , x_k ] := [[. . . [[x_1 , x_2 ], x_3 ], . . .], x_k ]$) is trivial for all $g \in G$. The result was generalised by E. Khukhro and P. Shumyatsky in a 2013 paper via an analysis of the sets $$E_{G,k }(x) := \{[g,_k x] : g \in G\}.$$ In this talk, we will continue to study the properties of these sets, concluding with the proof of two conjectures made in said paper. As a by-product of our methods, we also prove a generalisation of a result of Flavell, which itself generalises Wielandt’s Zipper Lemma and provides a characterisation of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism. We will finish the talk with some related work on the question: Which finite groups $G$ can have an element contained in a unique maximal subgroup of $G$? All of this is joint work with R. M. Guralnick.

For some classes of groups, there is a natural notion of rank, which can be used to argue by induction or sometimes even classify the groups: for example, the order of a finite group, or the dimension of a Lie group. Closely related is the pervasive theme of decomposing a group into "basic" or "irreducible" factors. How far can we get with this approach in the class of totally disconnected locally compact second-countable (t.d.l.c.s.c.) groups? I will describe a certain approach to structural complexity of t.d.l.c.s.c. groups that is inspired by developments in the area over the last ten years, particularly the class of elementary groups introduced P. Wesolek in his 2014 PhD thesis. The latter work shows that one can get a surprising amount of information from descending chain conditions on subgroups, and associated ordinal-valued rank functions, from a perspective that takes all compact groups and discrete groups as having small rank. I will give an example of a class of "well-founded" groups with good closure properties that properly contains the elementary groups, including for example all locally linear groups and many examples of compactly generated simple groups acting on trees with Tits' independence property, but then also give a family of t.d.l.c.s.c. groups that do not belong to this class.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

Groups acting on trees play a fundamental role in the theory of groups. Bass--Serre Theory, and in particular the notion of a graph of groups, is a powerful tool for decomposing groups acting on trees. It is also useful for constructing discrete groups acting on trees. However, its usefulness for constructing non-discrete groups acting on trees is, in some situations, severely limited. Such groups play an important role the theory of locally compact groups, as they are a rich source of examples of compactly generated simple groups. An alternative, but complementary, approach to the study of groups acting on trees has recently emerged based on local actions. Its origins can be traced back to the work of J.~Tits, but it was in a seminal paper of M.~Burger and Sh.~Mozes that this ``local-to-global'' approach was first fully articulated. Typically, these local-to-global constructions have something called Tits' independence property (P). Intuitively this property means that a group can act independently on different branches of the tree. In joint work with Colin Reid, we have developed a general method for describing and classifying all actions of groups on trees with property (P). This is done using an object called a local action diagram, akin to a graph of groups, but for local actions. Our work can be seen as a `local action' complement to Bass-Serre theory. As an example of how effective this local action diagram approach is, for a group $G$ with property (P) one can easily determine if $G$ is simple directly from its local action diagram.

We say that a finite group G acting on a set X has Property (*)_p for a prime p if the stabiliser of x in P is a Sylow p-subgroup of the stabiliser of x in G, for all x in X and Sylow p-subgroups P of G. Property (*)_p arose in the recent work of Tornier (2018) on local Sylow p-subgroups of Burger-Mozes groups, and he determined the values of p for which the alternating and symmetric groups in their natural actions have Property (*)_p. In this talk, we will explore the various properties of groups satisfying (*)_p and extensions of Tornier's result

Wielandt introduced the notion of the 2-closure of a permutation group $G$ on a set $\Omega$. This is the largest subgroup of $\mathrm{Sym}(\Omega)$ with the same set of orbits on ordered pairs as $G$. We say that $G$ is 2-closed if $G$ is equal to its 2-closure. The automorphism group of a graph or digraph is a 2-closed group. In this talk I will discuss some recent work with Luke Morgan and Jin-Xin Zhou on 2-closed groups that are not the automorphism group of a graph or digraph.

We give a full characterisation, in term of symmetries of the defining Coxeter system, of finitely generated Coxeter groups for which the group of automorphisms of the Cayley graph (with respect to the standard generating set) is uncountable and therefore non-discrete with the permutation topology. I will sketch the main ideas of the proof and, time permitting, I will mention results on rigidity. (Joint work with Federico Berlai)

The aim of this talk is to introduce the Euler-Poincaré characteristic in the context of totally disconnected locally compact (= TDLC) groups. For discrete groups, such a characteristic is just an integer or a rational number but, for TDLC-groups, it becomes a rational multiple of a Haar measure. This important invariant is also (mysteriously) related to the value in -1 of a double-coset zeta function that can be attached to a TDLC-group whenever a compact open subgroup is selected. We will discuss the definition of this type of zeta-function in detail. Joint work with Gianmarco Chinello and Thomas Weigel.

The notion of an iterated monodromy group, introduced by Nekrashevych, is a natural extension of the classical monodromy group of a covering. A particularly interesting source of examples comes from post-critically finite rational/polynomial maps. In this talk, we will recall the necessary definitions, along with a few well known examples, and then present a treatment of the class of conservative polynomials, introduced by Smale in his work on the Fundamental Theorem of Algebra. Some of the iterated monodoromy groups of conservative polynomials are finitely generated, dense subgroups in iterated wreath products of finite alternating groups and are branching over themselves (that is, as abstract groups, they are finitely generated permutational wreath products of themselves with an alternating group, G=Alt(d)xx(GxGx...xG)), while the others are branching over a subgroup of index 2 (a parity issue related to the multiplicities of the critical points of the polynomial).

For a tdlc group G and an endomorphism $\alpha$, the scale of $\alpha$ is $s(\alpha)=min\{[\alpha(U): U \cap \alpha(U) : U \text{ is a a compact open subgroup of } G\}$. In this talk, we will discuss an algorithm for computing the scale when the group is Neretin's group and the endomorphism is given by conjugation. The talk is based on ongoing work with Michal Ferov and George Willis.

The almost automorphism group of a regular tree is one of the most important examples in the theory of totally disconnected, locally compact groups. In this talk, we explain how to determine whether two of its elements are conjugate or not, combining results by Belk--Matucci and Gawron--Nekrashevych--Sushchanskii. This is joint work with Gil Goffer from the Weizmann Institute.

We explain a proof that Neretin groups have no nontrivial ergodic invariant random subgroups (IRS). Equivalently, any non-trivial ergodic p.m.p. action of Neretin’s group is essentially free. This property can be thought of as simplicity in the sense of measurable dynamics; while Neretin groups were known to be abstractly simple by a result of Kapoudjian. The heart of the proof is a commutator lemma for IRSs of elliptic subgroups.

Given a permutation group $G\leq \mathrm{Sym}(\Omega)$, a base for $G$ is a subset $B$ of Omega with trivial pointwise stabiliser. The cardinality of a minimal base for $G$ is called the base size and denoted $b(G)$. Base sizes of primitive permutation groups have attracted considerable interest in the literature over the past few decades, in part due to their tendency to have small base sizes in general. This talk will outline some recent developments in the study of base sizes of primitive affine groups, as well as some applications.

Let $P_k$ be the power map $x\mapsto x^k$ on a group $G$. We consider groups for which $P_k$ has dense image or $P_k$ is surjective. We study the structure such groups via linear representations using scale function and distality apart from general results from algebraic groups/linear algebra/tdlc groups.

In group theory, interesting statements about a group usually cannot be expressed in the language of first-order logic. It turns out, however, that some groups can actually be determined by their first-order properties, or, even more strongly, by a single first-order sentence. In the latter case the group is said to be finitely axiomatizable. I will describe some examples of this phenomenon (joint work with A. Nies and K. Tent). One family of results concerns axiomatizability of p-adic analytic pro-p groups, within the class of all profinite groups. Another main result is that for an adjoint simple Chevalley group of rank at least 2 and an integral domain R; the group G(R) is bi-interpretable with the ring R: This means in particular that first-order properties of the group G(R) correspond to first-order properties of the ring R. As many rings are known to be finitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every finitely generated group of the form G(R).

Based on an earlier work of Shelah concerning the relationship of the continuum hypothesis to the cardinality of the set of automorphisms of $\mathcal{P}(\omega)/\mathrm{fin}$, Velickovic showed that if such an automorphism admits a Borel lift $\mathcal{P}(\omega)\to \mathcal{P}(\omega)$, then it is of a certain "trivial" form. Similarly, Kanovei and Reeken showed that if $N,M$ are countable dense subgroups of $\mathbb{R}$, then every homomorphism $\mathbb{R}/N\to \mathbb{R}/M$ with a Borel lift $\mathbb{R}\to \mathbb{R}$, is of a certain "trivial" form. Kanovei and Reeken asked whether quotients of the $p$-adic groups satisfy similar "Ulam stability" phenomena. In this talk, we will settle this question by providing Ulam-stability phenomena for definable homomorphisms $G/N\to H/M$ when $G,H$ are arbitrary abelian non-archimedean Polish groups and $N,M$ are Polishable subgroups. This is joint work with Jeffrey Bergfalk and Martino Lupini.

I will discuss joint woIrk with Z. Ghadernezhad, where we show that under certain conditions the standard topology on the automorphism of a countable homogeneous structure (generated by the fixed point stabilizers of finite sets) is minimal among the Hausdorff group topologies on the group. I will also touch upon related results in the context of (generalized) Urysohn spaces.

We prove a local-to-global result for fixed points of groups acting on certain 2-dimensional affine buildings. Our proof combines very general CAT(0) space techniques with building-theoretic arguments. This is joint work with Jeroen Schillewaert and Koen Struyve.

Given a binary relation on the elements of a group, it is natural to study the properties of the graph encoding this relation. A well-known example is the generating graph, whose vertices are the non-identity elements of the group, and whose edges are its generating pairs. Earlier this year, Burness, Guralnick and Harper showed that if the generating graph of a finite group has no isolated vertices, then it as "dense" as possible, in the sense that it is connected with diameter at most 2. This generalises a famous result of Breuer, Guralnick and Kantor from 2008: the generating graph of a non-abelian finite simple group is connected with diameter 2. Consider now the non-commuting, non-generating graph of a group, obtained by taking the complement of the generating graph, removing edges between elements that commute, and finally removing all vertices corresponding to central elements. We explore the connectedness and diameter of this graph for various families of (finite and infinite) groups. In particular, we obtain a result that is perhaps surprising: in many cases, this naturally defined subgraph of the complement of the dense generating graph is itself similarly dense.

We say a group is invariably generated by a subset if every tuple in the product of conjugacy classes of elements in that subset is a generating tuple. We discuss the history of computational Galois theory and probabilistic generation problems to motivate some results about the probability of generating invariably a finite simple group, joint work with Daniele Garzoni. We also highlight some methods for studying probabilistic invariable generation.

Let $G$ be a finite primitive permutation group acting on a set $X$ with nontrivial point stabiliser $G_x$. We say that $G$ is extremely primitive if $G_x$ acts primitively on every orbit in $X\setminus\{x\}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. After surveying previous results, we will discuss joint work with Tim Burness towards completing this classification dealing with the almost simple groups with socle an exceptional group of Lie type. We will describe the various techniques used in the proof and, discuss the results we proved on bases for primitive actions of exceptional groups.

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces, we determine all irreducible modules for simple algebraic groups that are self-dual and have finitely many orbits on totally singular $k$-spaces ($k=1$ or $k=2$). This question is naturally connected with the problem of finding for which pairs of subgroups $H$, $J$ of an algebraic group $G$ there are finitely many $(H,J)$-double cosets. We provide a solution to the question when $J$ is a maximal parabolic subgroup $P_k$ of a classical group.

(jt with Adam Thomas) The classical Jacobson–Morozov theorem guarantees that any nilpotent element $e$ in a semisimple complex Lie algebra $\mathfrak g$ can be extended to an $sl_2$-triple $(e,h,f)$ with $[h,e]=2e$, $[h,f]=-2f$ and $[e,f]=h$. This is a very useful theorem—for example in defining a Slodowy slice. A theorem of Kostant tells you the $sl_2$-triple is even unique up to conjugacy by the simple complex algebraic group $G$ with $\mathfrak g=\rm{Lie}(G)$. Building on previous work of Pommerening, Carter and others, Thomas and I gave precise conditions on the odd characteristic for these results to hold. The appropriate analogue in characteristic $2$ is subtle since an $sl_2$-triple generates a (nilpotent) Heisenberg algebra; one can also consider a $pgl_2$-triple with $[h,e]=e$, $[h,f]=f$ and $[e,f]=0$ having a $2$-dimensional abelian ideal; lastly, in characteristic $2$ there is a simple $3$-dimensional Lie algebra with $[e,f]=h$, $[h,e]=e$ and $[h,f]=f$—‘fake $sl_2$’. We give complete answers on the embeddings of $e$ into such subalgebras in all cases. An interesting waypoint is to classify the nilpotent elements admitting toral elements $h$ with $[h,e]=e$, in other words, to find the dimension of $$n_{\mathfrak g}({\rm span}(e))/c_{\mathfrak g}(e),$$ which is an interesting problem only in characteristic $2$.

The general theory of totally disconnected locally compact groups has advanced by leaps and bounds in recent years. Yet there is still need for examples of such groups that are simple and compactly generated. One such is the group of almost automorphisms of a rooted regular tree (a.k.a. Neretin's group). This turns out to also be an example of a piecewise full group (a.k.a topological full group) of homeomorphisms of the Cantor set (the boundary of the tree). These piecewise full groups have been a source of new examples of finitely generated infinite simple groups. I will report on ongoing joint work with Colin Reid and David Robertson on how to use piecewise full groups to obtain examples of compactly generated, simple, totally disconnected locally compact groups.

We will present the theory behind and new results on the cohomology of super-reflexive Banach G-modules X, where G is a countable discrete group. In particular, we shall show how the cohomology is controlled by the FC-centre of G, that is, the subgroup of elements having finite conjugacy classes. For example, using purely cohomological tools, we show that when X is an isometric super-reflexive Banach G-module so that X has no almost invariant unit vectors under the action of the FC-centre, then the associated cochain complex is split exact. Further connections to the work of Bader-Furman-Gelander-Monod, Nowak, and Bader-Rosendal-Sauer will be presented. We aim to start out slowly so that the talk should be accessible to the general analyst, geometer or group theorist.

First, we recall the definition of self-replicating groups acting on regular rooted trees and give examples. We then point at work of Baumgartner-Willis and a recent result of Willis which together highlight the greater significance of self-replicating groups in the theory of t.d.l.c. groups. Finally we outline the capabilities of and further plans for a GAP package for self-replicating groups. This package is being developed in joint work with students S. King and S. Shotter.

Countably infinite groups such as $\mathbb Z, \mathbb Q$, and $SL_n(\mathbb Q)$ are computable in the sense that they have a copy with domain the natural numbers and the group operations computable functions. (Such a definition works for any countable structure in a finite signature, not only for groups.) For f.g. groups, computability in this sense is equivalent to having a decidable word problem. \\ Most topological groups $G$ that have been studied are uncountable. In this case, one can attempt to define computability of $G$ via a computable structure of approximations to the element of $G$. We introduce various definitions of computability for t.d.l.c. groups $G$ and show their equivalence. One approach is to view $G$ as embedded into $S_\infty$ and use certain finite injections as approximations. Equivalently, one can take the ordered groupoid of compact open cosets of $G$ as the structure. Common t.d.l.c. groups such as $\mathrm{Aut}(T_d)$ and $SL_n(\mathbb Q_p)$ turn out to be computable in our sense. \\ In the abelian case, we address the question to which extent Pontryagin duality preserves the computability of groups. \\ This is joint work with A. Melnikov and also M. Lupini. \\ References: (1) Stone-type duality for totally disconnected locally compact groups, Logic Blog 2020, Section 4, arxiv.org/pdf/2101.09508.pdf (2) Computable topological abelian groups, with Melnikov and Lupini, arxiv.org/pdf/2105.12897.pdf

For a locally compact metrizable group $G$, we consider the action of $\text{Aut}(G)$ on Sub$_G$, the space of all closed subgroups of $G$ endowed with the Chabauty topology. We study the structure of groups $G$ admitting automorphisms $T$ which act expansively on Sub$_G$ and also the structure of contraction subgroups of such $T$. We show that such a group $G$ is necessarily totally disconnected and non-compact (unless it is finite) and that $T$ itself is expansive. Conversely, if $G$ is a $p$-adic field, then any expansive automorphism of $G$ acts expansively on Sub$_G$. However, higher dimensional $p$-adic vector spaces do not admit such automorphisms. (Joint work with Manoj B. Prajapati, Monatshefte f\"ur Mathematik (2020) 193:129--142 https://doi.org/10.1007/s00605-020-01389-5).

Many interesting and surprising results have arisen from studying generating sets for groups. For example, every finite simple group has a generating pair, and, moreover, every nontrivial element is contained in a generating pair. I will discuss recent work with Burness and Guralnick that completely classifies the finite groups where every nontrivial element is contained in a generating pair and answers a 1975 question of Brenner and Wiegold. I will explain how this generation problem is related to interesting questions about subgroup structure and how these questions can be addressed via the technique of Shintani descent.

Let G be a finite primitive permutation group on a set X with point stabiliser H and recall that a subset of X is a base if its pointwise stabiliser is trivial. The base size of G, denoted b(G), is the minimal size of a base. In this talk, I will present several new results that give bounds on b(G) under various structural restrictions on H. For example, a theorem of Seress from 1996 states that if G is soluble then b(G) is at most 4 and I have recently proved that b(G) is at most 5 if one only assumes that H is soluble (both bounds are best possible). I will report on some natural extensions in joint work with Aner Shalev and time permitting, I will present new results with Hongyi Huang on the Saxl graphs of base-two primitive groups with soluble stabilisers.

Groups acting on rooted trees, especially the so-called branch groups, have been vastly studied over the past few decades, owing to their exotic properties - in particular, branch groups have been used to answer important open problems and disprove conjectures. The study of maximal subgroups of branch groups has recently picked up speed, with new developments by Francoeur enabling one to study the maximal subgroups of the larger class of weakly branch groups. A prominent example of a weakly branch, but not branch, group is the Basilica group. This was the first example of an amenable group which is not subexponentially amenable. In this talk, I will present results concerning maximal subgroups of a family of generalised Basilica groups. This is joint work with Karthika Rajeev.

In this talk we will discuss the structure of maximal subgroups of finite simple groups, particularly groups of Lie type. We will discuss subgroups of exceptional groups of Lie type, and a version of Ennola duality that exists for groups of Lie type, which relates untwisted and twisted groups of Lie type.