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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.

Florian Breuer: Realizing general linear groups as Galois groups

23rd November, 2020

I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed.

William Hautekiet: Automorphism groups of transcendental field extensions

23rd November, 2020

It is well-known that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made.

Wenhao Wang: Dehn Functions of Finitely Presented Metabelian Groups

12th November, 2020

The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group G is 2n2k, where k is the torsion-free rank of the abelianization Gab, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.

Marcos Mazari-Armida: Model-theoretic stability in classes of modules

12th November, 2020

Dividing lines in complete first-order theories were introduced by Shelah in the early seventies. A dividing line is a property such that the classes satisfying such a property have some nice behaviour while those not satisfying it have a bad one. Two of the best understood dividing lines are those of stability and superstability.

In this talk, I will study the notion of stability and superstability in abstract elementary classes of modules with respect to pure embeddings, i.e., classes of the form (K,≤p) where K is a class of R-modules for a fixed ring R and ≤p is the pure submodule relation. In particular, using that the class of p-groups with pure embeddings is a stable AEC, I will present a solution to Problem 5.1 in page 181 of Abelian Groups by László Fuchs. Moreover, I will show how the notion of superstability can be used to give new characterizations of noetherian rings, pure-semisimple rings, and perfect rings.

Henry Bradford: Quantitative LEF and topological full groups

9th November, 2020

Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore some ways in which asymptotic properties of the finite groups which occur reflect asymptotic properties of the associated subshift.

Olga Kharlampovich: Fraïssé limits of limit groups

29th October, 2020

We modify the notion of a Fraïssé class and show that various interesting classes of groups, notably the class of non-abelian limit groups and the class of finitely generated elementary free groups, admit Fraïssé limits. We will also discuss countable elementary free groups.

Rachel Skipper: Maximal Subgroups of Thompson’s group V

16th October, 2020

There has been a long interest in embedding and non-embedding results for groups in the Thompson family. One way to get at results of this form is to classify maximal subgroups. In this talk, we will define certain labellings of binary trees and use them to produce a large family of new maximal subgroups of Thompson's group V. We also relate them to a conjecture about Thompson's group T.

Benjamin Steinberg: Simplicity of Nekrashevych algebras of contracting self-similar groups

8th October, 2020

A self-similar group is a group G acting on the Cayley graph of a finitely generated free monoid X* (i.e., a regular rooted tree) by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups like Grigorchuk's 2-group of intermediate growth are of this form. Nekrashevych associated C*-algebras and algebras with coefficients in a field to self-similar groups. In the case G is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk we discuss a recent result of the speaker and N. Szakacs (York/Szeged) characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like Gupta-Sidki groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

Laura Ciobanu: On computing fixed subgroups of endomorphisms in free groups

1st October, 2020

Given an endomorphism h of a free group F, the fixed subgroup of h consists of those elements x ∈ F for which h(x)=x. In this talk I will give some background on fixed subgroups in free groups, and then present an algorithm which computes the fixed subgroup and the stable image for any endomorphism of the free group of rank 2. This answers, for rank 2, a question posed by Stallings in 1984 and a more recent question of Ventura. I will explain why general endomorphisms are more difficult than automorphisms, and in what ways our algorithm needs the restriction on the rank. This is joint work with Alan Logan.

Ana Khukhro: A new characterization of virtually free groups

4th September, 2020

A finite graph that can be obtained from a given graph by contracting edges and removing vertices and edges is said to be a minor of this graph. Minors have played an important role in graph theory, ever since the well-known result of Kuratowski that characterized planar graphs as those that do not admit the complete graph on five vertices nor the complete bipartite graph on (3,3) vertices as minors. In this talk, we will explore how this concept interacts with some notions from geometric group theory, and describe a new characterisation of virtually free groups in terms of minors of their Cayley graphs.

Murray Elder: Rewriting systems and geodetic graphs

4th September, 2020

I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof relies on a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.

Simon Smith: Infinite primitive permutation groups, cartesian decompositions, and topologically simple locally compact groups

21st August, 2020

A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just-infinite groups. In this talk, I'll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class 𝒮 of non-discrete, topologically simple, compactly generated, tdlc groups. This class 𝒮 plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in 𝒮 are known.

Tony Guttmann: On the amenability of Thompson’s Group F

7th August, 2020

In 1967 Richard Thompson introduced the group F, hoping that it was non-amenable, since then it would disprove the von Neumann conjecture. Though the conjecture has subsequently been disproved, the question of the amenability of Thompson's group F has still not been rigorously settled. In this talk I will present the most comprehensive numerical attack on this problem that has yet been mounted. I will first give a history of the problem, including mention of the many incorrect "proofs" of amenability or non-amenability. Then I will give details of a new, efficient algorithm for obtaining terms of the co-growth sequence. Finally I will describe a number of numerical methods to analyse the co-growth sequences of a number of infinite, finitely-generated groups, and show how these methods provide compelling evidence (though of course not a proof) that Thompson's group F is not amenable. I will also describe an alternative route to a rigorous proof.

Collin Bleak: On the complexity of elementary amenable subgroups of R. Thompson’s group F

7th August, 2020

The theory of EG, the class of elementary amenable groups, has developed steadily since the class was introduced constructively by Day in 1957. At that time, it was unclear whether or not EG was equal to the class AG of all amenable groups. Highlights of this development certainly include Chou's article in 1980 which develops much of the basic structure theory of the class EG, and Grigorchuk's 1985 result showing that the first Grigorchuk group G is amenable but not elementary amenable. In this talk we report on work where we demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group F which is strictly well-ordered by the embeddability relation in type ε0 + 1. All except the maximum element of this family (which is F itself) are elementary amenable groups. In this way, for each α less than ε0, we obtain a finitely generated elementary amenable subgroup of F whose EA-class is α + 2. The talk will is pitched for an algebraically inclined audience, but little background knowledge will be assumed.

Harry Hyungryul Baik: Normal generators for mapping class groups are abundant in the fibered cone

17th July, 2020

We show that for almost all primitive integral cohomology classes in the fibred cone of a closed fibred hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fibre. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analogue of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex.

Hiroki Matui: Various examples of topological full groups

26th June, 2020

I will begin with the definition of topological full groups and explain various examples of them. The topological full group arising from a minimal homeomorphism on a Cantor set gave the first example of finitely generated simple groups that are amenable and infinite. The topological full groups of one-sided shifts of finite type are viewed as generalization of the Higman-Thompson groups. Based on these two fundamental examples, I will discuss recent development of the study around topological full groups.

Tianyi Zheng: Neretin groups admit no non-trivial invariant random subgroups

26th June, 2020

We explain the proof that Neretin groups have no non-trivial 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 'double commutator' lemma for IRSs of elliptic subgroups.

Mark Hagen: Hierarchical hyperbolicity from actions on simplicial complexes

5th June, 2020

The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition - which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes - is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups.