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Sven Raum: Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

9th August, 2021

In the 90s, Nebbia conjectured that a group of tree automorphisms acting transitively on the tree's boundary must be of type I, that is, its unitary representations can in principal be classified. For key examples, such as Burger-Mozes groups, this conjecture is verified. Aiming for a better understanding of Nebbia's conjecture and a better understanding of representation theory of groups acting on trees, it is natural to ask whether there is a characterization of type I groups acting on trees. In 2016, we introduced in collaboration with Cyril Houdayer a refinement of Nebbia's conjecture to a trichotomy, opposing type I groups with groups whose von Neumann algebra is non-amenable. For large classes of groups, including Burger-Mozes groups, we could verify this trichotomy.

In this talk, I will motivate and introduce the conjecture trichotomy for groups acting on tress and explain how von Neumann algebraic techniques enter the picture.

Lancelot Semal: Unitary representations of totally disconnected locally compact groups satisfying Ol’shanskii’s factorization

5th June, 2021

We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings.

Yago Antolin: Geometry and Complexity of positive cones in groups

10th May, 2021

A positive cone on a group G is a subsemigroup P such that G is the disjoint union of P, P−1 and the trivial element. Positive cones codify naturally G-left-invariant total orders on G. When G is a finitely generated group, we will discuss whether or not a positive cone can be described by a regular language over the generators and how the ambient geometry of G influences the geometry of a positive cone.

Ilya Kapovich: Non-linear words and free groups

29th April, 2021

An important theme in the study of combinatorics of words involves looking for models of nonlinear words, that is words that are not indexed by segments of integers. We discuss one such model arising from the theory of Stallings subgroup graphs. This model naturally leads to the notion of subset currents on free groups (and on other word-hyperbolic groups) which are measure-theoretic analogs of conjugacy classes of finitely generated subgroups. Many new features manifest themselves in this context, including connections with the Hanna Neumann Conjecture and Whitehead's algorithm for subgroups.

Dawid Kielak: Recognizing surface groups

8th March, 2021

I will address two problems about recognizing surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension 2. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognizing surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.

Charlotte Hoffmann: Short words of high imprimitivity rank yield hyperbolic one-relator groups

8th March, 2021

It is a long-standing question whether a group of type F that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type F and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than 2, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most 17. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I'll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.

Paul-Henry Leemann: Cayley graphs with few automorphisms

22nd February, 2021

Let G be a group and S a generating set. Then the group G naturally acts on the Cayley graph Cay(G,S) by left multiplications. The group G is said to be rigid if there exists an S such that the only automorphisms of Cay(G,S) are the ones coming from the action of G. While the classification of finite rigid groups was achieved in 1981, few results were known about infinite groups. In a recent work, with M. de la Salle we gave a complete classification of infinite finitely generated rigid groups. As a consequence, we also obtain that every finitely generated group admits a Cayley graph with countable automorphism group.

Alina Vdovina: Buildings, quaternions and Drinfeld-Manin solutions of Yang-Baxter equations

11th February, 2021

We will give a brief introduction to the theory of buildings and present their geometric, algebraic and arithmetic aspects. In particular, we present explicit constructions of infinite families of quaternionic cube complexes, covered by buildings. We will introduce new connections of geometric group theory and theoretical physics by using quaternionic lattices to find new infinite families of Drinfeld-Manin solutions of Yang-Baxter equations.

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.

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.

Anna Erschler: Ordering Ratio Function and Travelling Salesman Breakpoint for Groups and Metric Spaces

4th December, 2020

(Abstract taken from arXiv paper.) We study asymptotic invariants of metric spaces and infinite groups related to the universal Travelling Salesman Problem (TSP). We prove that spaces with doubling property (in particular virtually nilpotent groups) admit Gap for Ordering Ratio functions which holds for all orders on these spaces. We describe Travelling Salesman Breakpoint for finite graphs. We characterize groups with Travelling Salesman Breakpoint ≤ 3 as virtually free ones. We show that Ordering Ratio function is bounded (which is the best possible situation) for all uniformly discrete δ-hyperbolic spaces of bounded geometry, in particular for all hyperbolic groups. We prove that any metric space, containing weakly a sequence of arbitrarily large cubes, has infinite Travelling Salesman Breakpoint; this means that any order on such spaces satisfies OR(s)=s for all s. This is the worst possible case for Ordering Ratio functions. For a sequence of finite graphs, we provide a sufficient spectral condition for OR(s)=s. This condition is in particular satisfied for any sequence of expander graphs. Under this stronger assumption of being a family of expander graphs, we prove a stronger claim about snakes of bounded width. We show that any metric space of finite Assouad-Nagata dimension admits an order satisfying OR(s) ≤ Const ln s, and discuss general Gap Problems for Ordering Ratio functions.

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.

George Domat: Free products from spinning and rotating families

19th November, 2020

A natural goal of geometric group theory is to understand the algebraic properties of a group via geometry. The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup in a large ambient group. In particular, their work gives conditions under which the normal closure is a free product. I will talk about recent work that aims to unify their results and gives a significantly shorter proof of the theorem of DGO. This is joint work with M. Bestvina, R. Dickmann, S. Kwak, P. Patel, and E. Stark.

Doron Puder: Random permutations sampled by free words

16th November, 2020

Fix a word w in a free group on r generators. A w-random permutation in the symmetric group SN is obtained by sampling r independent uniformly random permutations σ1, . . .,σr ∈ SN and evaluating w(σ1, . . .,σr). Such w-random permutations have surprisingly rich structure with relation to deep results in geometric group theory. I'll survey some of this structure, state some conjectures, and explain how it is related to evaluating the spectral gap of random Schreier graphs of SN.

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.

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.

Abdul Zalloum: Regularity of Morse geodesics and growth of stable subgroups

17th September, 2020

The study of groups with "hyperbolic-like directions" has been a central theme in geometric group theory. Two notions are usually used to quantify what is meant by "hyperbolic-like directions", the notion of a contracting geodesic and that of a Morse geodesic. Since the property that every geodesic ray in metric space is contracting or Morse characterizes hyperbolic spaces, being a contracting/Morse geodesic is considered a hyperbolic-like property. Generalizing work of Cannon, I will discuss a joint result with Eike proving that for any finitely generated group, the language of contracting geodesics with a fixed parameter is a regular language. This immediately implies that contracting geodesics can't exist in torsion groups.

The Morse notion is a weaker notion than that of the contracting notion, in fact, building on work of Osin, Ol’shanskii, and Sapir, Fink gave an example of a torsion group which contains an infinite Morse geodesic. This seems to contradict the claim that Morse geodesics are "hyperbolic-like" directions. As an attempt to rectify this, Russell, Spriano, and Tran introduced a class of spaces where Morse (quasi)-geodesics satisfy some local-to-global property and they showed that many interesting examples live in such a class. In these spaces, Morse (quasi)-geodesics are expected to behave more reasonably like "hyperbolic directions", therefore, such spaces/groups can be regarded as good hosts of Morse (quasi)-geodesics.

I will discuss some continuation of their work where we show that in such spaces Morse geodesics form a regular language, give a characterization of stable subgroups in terms of regular languages. Time permitting, I will discuss few other applications of these automatic structures to the growth of stable subgroups and the dynamics of the action of such groups on their Morse boundaries. This work is joint with Cordes, Russell and Spriano.

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.