Tag - Group theory

Charles Cox: Spread and infinite groups

25th January, 2021

My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups. A group G has spread k if for every g1,…,gk we can find an h in G such that ⟨gi,h⟩=G. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread 0. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven's first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we'll see the answer to this question. The arguments will be concrete and accessible to a general audience.

François Le Maître: Dense totipotent free subgroups of full groups

25th January, 2021

In this talk, we will be interested in measure-preserving actions of countable groups on standard probability spaces, and more precisely in the partitions of the space into orbits that they induce, also called measure-preserving equivalence relations. In 2000, Gaboriau obtained a characterization of the ergodic equivalence relations which come from non-free actions of the free group on n > 1 generators: these are exactly the equivalence relations of cost less than n. A natural question is: how non-free can these actions be made, and what does the action on each orbit look like? We will obtain a satisfactory answer by showing that the action on each orbit can be made totipotent, which roughly means 'as rich as possible', and furthermore that the free group can be made dense in the ambient full group of the equivalence relation.

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.

Eugene Plotkin: On logical rigidity of groups

4th December, 2020

We will survey a series of recent developments in the area of first-order descriptions of linear groups. The goal is to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups. We also dwell on the principal problem of isotipicity of finitely generated groups.

Henry Wilton: Negative immersions and one-relator groups

4th December, 2020

One-relator groups G=F/≪w≫ pose a challenge to geometric group theorists. On the one hand, they satisfy strong algebraic constraints (eg Magnus's theorem that the word problem is soluble). On the other hand, they are not susceptible to geometric techniques, since some of them (such as Baumslag-Solitar groups) exhibit extremely pathological behaviour.

I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder - the primitivity rank π(w), the smallest rank of a subgroup of F containing w as an imprimitive element. A sample application is that every subgroup of G of rank less than π(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups.

Agatha Atkarskaya: Small cancellation rings

4th December, 2020

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. Joint results with A. Kanel-Belov, E. Plotkin, E. Rips.