# Schedule

## Schedule February 2024 - July 2024

14/02/2024 - 15:00 CET

## Jun Ueki

### The p-adic limits of torsions in the Z-covers of knots

We investigate the p-adic limit values of the torsion sizes of the 1st homology groups in the Z-covers of knots.

We compare the results with the cases of elliptic curves and give a remark on an analogue of the Lang--Trotter conjecture (Elkies's theorem) for an infinite set of knots from the viewpoint of arithmetic topology.

This talk is based on joint works with Hyuga Yoshizaki and Sohei Tateno.

06/03/2024 - 14:00 CET

### Genericity properties in braid groups

Roughly speaking, a property in a group is said to be generic if "almost every" element in the group has this property. This concept has at least two different technical definitions and finds several applications. We will focus on the generic properties that have been studied in braid groups, namely being pseudo-Anosov and computational properties. Additionally, we will explore how these properties can impact the security of some braid-based cryptosystems.

27/03/2024 - 14:00 CET

### Diameter bounds for finite classical groups generated by special elements

The diameter of a group G with respect to a symmetric generating set X is the smallest integer d such that every element of G is the product of at most d elements of X. A well-known conjecture of Babai predicts that every nonabelian finite simple group G has diameter (log |G|)^O(1) with respect to any generating set. This is known to be true for bounded-rank groups of Lie type (Helfgott; Pyber--Szabo; Breuillard--Green--Tao), but the conjecture is wide open for high-rank groups. There has been a good deal of progress recently for generating sets containing either special elements or random elements. For example, we now know that Babai's conjecture holds for SL_n(q) for any generating set include a transvection, or for three random elements (q bounded). In this talk I will explain these results in detail and outline some of the main ideas that go into the proof. Some of these results are joint work with Jezernik, while others build on recent work of Garonzi--Halasi--Somlai.

17/04/2024 - 14:00 CEST

On the subgroup membership problem in bounded automata groups

The class of bounded automata groups includes many important examples of tree automorphism groups such as the Grigorchuk group and the Gupta-Sidki groups. Given a finite generating set X for a group G, the subgroup membership problem is then stated as follows: given a description of some subgroups H of G, compute a description of all the words over X which evaluate to elements of the subgroup H. Notice that the word problem is an instance of the subgroup membership problem. In the literature, subgroup membership problems have been considered for finitely generated subgroups.

We extend what is known by instead considering the infinitely generated subgroups of bounded automata groups which can be specified as the stabiliser of quasi-periodic rays. We show that for such subgroups, the subgroup membership problem (and its set complement) is an ET0L language, that this ET0L language is effectively constructible, and that membership to such subgroups is decidable.

The techniques used in this work have applications to the study of the word and coword problem of bounded automata groups.

This is joint work with Daniele D'Angeli, Francesco Matucci, Tatiana Nagnibeda, Davide Perego and Emanuele Rodaro.

22/05/2024 - 14:00 CEST

### Probabilistic laws on groups

Suppose a finite group satisfies the following property: If you take two random elements, then with probability bigger than 5/8 they commute. Then this group is commutative.

Starting from this well-known result, it is natural to ask: Do similar results hold for other laws (p-groups, nilpotent groups...)? Are there analogous results for infinite groups? Are there phenomena specific to the infinite setup?

We will survey known and new results in this area. New results are joint with Gideon Amir, Maria Gerasimova and Gady Kozma.

29/05/2024 - 14:00 CEST

### Solubilizers, nilpotentizers and p-elements in profinite groups

Let C be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group G and an element x in G, we are interested in the probability that a randomly chosen element of G generates a pro-C subgroup together with x.

For different choices of C, we will discuss the following questions: is there a characterization of the elements of G with the property that this probability is positive? what can be deduced about the structure of G if we know that this probability is positive for all the elements of G?

03/07/2024 - 14:00 CEST

### Dyer groups: Coxeter groups, right-angled Artin groups and more...

Dyer groups are a family encompassing both Coxeter groups and right-angled Artin groups. Indeed these two classes of groups share many properties: they have the same solution to the word problem, intersections of parabolic subgroups are parabolic, they are CAT(0)... So which of those generalize to Dyer groups? In this talk I will introduce Dyer groups and give some of their properties.

## Schedule October 2022 - February 2023

Tuesday 15/11/2022 - 16:00 CET

## Andoni Zozaya

### Degree of commutativity and wreath products

Abstract: Based on joint work with Iker de las Heras and Benjamin Klopsch, I will discuss results and open questions regarding the degree of commutativity in finitely generated (infinite) groups. Firstly, I will focus on the initial work of Antolín, Martino and Ventura that led to some results and to the main conjecture of whether any group with exponential growth rate has degree of commutativity equal to zero. I will recall some positive results and show the new evidence we produced; namely, every wreath product with infinite cyclic top group has degree of commutativity equal to zero.

Tuesday 06/12/2022 - 16:00 CET

## Matthew Tointon

### Transience of random walks on vertex-transitive graphs via growth and isoperimetry in groups

Abstract: A random walk on a graph is called recurrent if it eventually returns to its starting point with probability 1, and transient otherwise. Varopoulos famously showed that the simple random walk on an infinite vertex-transitive graph G is recurrent if and only if G is quasi-isometric to the standard Cayley graph of Z or Z^2. In this talk I will describe how recent quantitative work on growth and isoperimetry of groups, joint with Romain Tessera, allows us to prove a quantitative, finitary version of Varopoulos's theorem. Amongst other things this proves that there is a gap at 1 for the return probability of the simple random walk on a vertex-transitive graph, i.e. an absolute constant eps>0 such that if the probability that the walk eventually returns to its starting point is at least 1-eps then the walk is recurrent. It also verifies (indeed, strengthens) an analogue of Varopoulos's result for finite graphs conjectured by Benjamini and Kozma in 2002.

Tuesday 20/12/2022 - 16:00 CET

## Stefan Witzel

Strong property (T) for Ã_2-lattices

Abstract: Lafforgue introduced strong property (T) as a strengthening of Kazhdan's property (T) to non-unitary representations and proved that it is satisfied by higher-rank Lie groups and their uniform lattices. Strong property (T) for non-uniform higher-rank arithmetic lattices was proven by de la Salle and subsequently used in the proof of Zimmer's conjecture by Brown, Fisher, Hurtado.

I will speak about the proof of strong property (T) for lattices on Ã_2 buildings that may not be arithmetic. This is joint work with Mikael de la Salle and Jean Lécureux.

## Schedule March - July 2022

Thursday 03/03/2022 - 17:00 CET

## Dominik Francoeur

### The quasi-transitivity degree of branch groups

Abstract: The action of a group G on a set X is said to be quasi-k-transitive if the diagonal action of G on k copies of X has only finitely many orbits. This is a natural generalisation of the notion of k-transitivity. In this talk, we will be interested in understanding the possible quasi-k-transitive actions for a family of groups known as branch groups, which are a special kind of groups acting on rooted trees with interesting properties.

Friday 26/03/2022 - 16:00 CET

## Davide Spriano

### Hyperbolic spaces for CAT(0) groups

Abstract: A common theme in geometric group theory is to find interesting hyperbolic spaces on which a group acts. Examples of this are the action of the Mapping Class Group on the curve graph or the action of Out(F_n) on the free splitting complex. In this talk, we will construct of hyperbolic spaces for groups acting geometrically on CAT(0) spaces. The construction is inspired by the contact graph (and generalizations) for CAT(0) cube complexes, but there are complications as general CAT(0) spaces lack the extra combinatorial data given by cube complexes. As an application, we obtain description of rank-one elements and use rank-rigidity type of results. This is joint work with H. Petyt and A. Zalloum.

Friday 06/05/2022 - 16:00 CEST

## Marco Moraschini

### New computations in bounded cohomology

Bounded cohomology is a functional-analytic variant of ordinary cohomology. Despite its many applications in geometric topology and group theory, it is notoriously hard to compute bounded cohomology. For instance, we only know the bounded cohomology of the non-Abelian free group of rank 2 up to degree 3. On the other hand, it is well known that bounded cohomology vanishes in all positive degrees (and real coefficients endowed with the trivial action) in the presence of amenable groups. However, finding other instances of this situation is more complicated.

In this talk we will report on some recent computations in bounded cohomology with special emphasis on the case of boundedly acyclic groups, i.e. groups with vanishing bounded cohomology in all positive degrees (and trivial real coefficients).

This is a joint work with Francesco Fournier-Facio and Clara Löh.

Friday 10/06/2022 - 16:00 CEST

## Anna Giordano Bruno

### Growth and entropy for group endomorphisms

We extend the classical notion of growth rate of finitely generated groups to group endomorphisms, and we define when a group endomorphism has polynomial, exponential or intermediate growth. In the spirit of Gromov’s Theorem and Chou’s extension of Milnor-Wolf’s Theorem, we see that the endomorphisms of elementary amenable groups have either polynomial or exponential growth. The same dichotomy holds for the endomorphisms of locally finite groups.

The growth of group endomorphisms is strictly connected to the algebraic entropy, an invariant for group endomorphisms inspired by the notions of metric entropy and topological entropy in ergodic theory and topological dynamics. It is known that the algebraic entropy is an additive invariant in the Abelian case, while an example from the classical theory of growth of finitely generated groups shows that this is not the case in general, even for solvable groups. On the other hand, we conjecture the additivity of the algebraic entropy for the endomorphisms of nilpotent groups and of locally finite groups; in the latter case we prove it under some additional hypotheses.

This talk is based on joint works with Dikran Dikranjan, Flavio Salizzoni, Pablo Spiga, Simone Virili.

Friday 17/06/2022 - 16:00 CEST

## Balint Virag

### Amenability of quadratic automata groups

Polynomial automata groups are exciting examples but many basic properties are still open. I will review what these groups are and a new result that quadratic ones are amenable. Joint with Amir and Angel.

Friday 08/07/2022 - 16:00 CEST

## Ángel del Río

### The Isomorphism Problem for group rings

Let $R$ be a ring. The Isomorphism Problem for group rings over $R$ asks whether the isomorphism type of a group $G$ is determined by the isomorphism type of the group ring $RG$.

We will review the main results on the Isomorphism Problem for group rings of finite groups. The history of the Isomorphism Problem goes back to a seminal paper of G. Higman in the 1940s who proved that $G$ and $H$ are two non-isomorphic groups and one of them is abelian then the integral group rings $\mathbb{Z}G and $\mathbb{Z}H$ are not isomorphic. This was generalized to $G$ metabelian by Withcomb in 1950 and to $G$ abelian-by-nilpotent by Roggenkamp and Scott in the late 1980's.

However, in 2000, Hertweck showed two non-isomorphic solvable groups with isomorphic integral group rings.

Previously, Dade had showed in 1972 two non-isomorphic metabelian groups $G$ and $H$ with $FG$ and $FH$ isomorphic for every field $F$.

The special case of the Isomorphism Problem where $R$ is a field with $p$ elements and $G$ is a finite $p$-group, for $p$ prime, is known as the Modular Isomorphism Problem.

It appeared for the first time in a survey paper by R. Brauer in 1963 and was open in general until recently.

In cooperation with Diego García and Leo Margolis we discovered two non-isomorphic groups of order $2^9$ whose group algebras over any field of characteristic $2$ are isomorphic.

However no counterexample for $p$ odd is known. We will present the counterexample and some recent results for $p$ odd.

## Schedule May - July 2021

Wednesday 19/05/2021 - 17:00 CEST

## Ndeye Coumba Sarr

### Almost amalgamated profinite groups

Abstract: Bass-Serre theory was initiated in the 1970 by J.P. Serre and aims to characterize the properties of certain groups (freedom or the fact of being an amalgam of groups) by making them act on trees. One of the fundamental results of this theory states that a group is free if and only if it acts freely on a tree.

In 2011, B. Deschamps and I. Suarez introduced a profinite version of this result and lay the foundations of a profinite Bass-Serre theory. More precisely they have been interested in an analogue for profinite groups of this following Serre's result: an abstract group is free if and only if it acts freely on a tree.

In this talk we will first introduce Deschamp-Suarez prographs theory, then we will prove a profinite analogue of Serre's combinatorial caracterisation of amalgamted groups.This lead us to consider the notion of almost amalgamated profinite groups, i.e. which contain a dense amalgamated abstract subgroup. We will finish with an illustration of this result in Galois theory: we will describe the absolute Galois group ofR((t)) (isomorphic to Dp∞=Zp \rtimes Z/2Z) by using the arithmetic of this field.

Wednesday 02/06/2021 - 17:00 CEST

## Gabor Elek

### Uniform amenability

Abstract: According to the classical result of Connes, Feldman and Weiss, measured hyperfiniteness of a group action is equivalent to measured amenability. In the Borel category it is known that hyperfiniteness implies amenability and it is conjectured that the converse is true. Based on the work of Anantharaman-Delaroche and Renault, one can introduce the notion of uniform amenability, a strengthening of measured amenability (it is a sort of exactness in the category of measurable actions, so the famous Gromov-Osajda groups have no free uniformly amenable actions). One can also introduce the notion of uniform hyperfiniteness in a rather natural way. We prove that the two notions are equivalent provided that the measurable action satisfies a boundedness condition for the Radon-Nikodym derivative (e.g. in the case of Poisson boundaries).

Wednesday 09/06/2021 - 17:00 CEST

## Federico Berlai

### Automorphism groups of Cayley graphs of Coxeter groups

Abstract: It is known that automorphism groups of locally finite graphs admit a totally disconnected locally compact (tdlc) topology. In this talk I will present some recent results concerning automorphism groups of a particular class of locally finite graphs, that is of Cayley graphs of Coxeter groups. Particular attention will be given to the right-angled case.

Joint work with Michal Ferov.

Wednesday 16/06/2021 - 17:00 CEST

## Jan Moritz Petschick

### The Basilica Operation

Abstract: The Basilica group is a well-studied example of a group of automorphisms of the dyadic rooted tree. We will explore the connection between it and the binary odometer, and derive a construction that allows us to associate a family of Basilica groups to every group of automorphisms of a rooted regular tree. We consider the inheritance properties of this construction, and apply this to calculate the Hausdorff dimension of some spinal groups. This is joint work with Karthika Rajeev.

Wednesday 23/06/2021 - 17:00 CEST

## Giles Gardam

### Kaplansky's conjectures

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.

Wednesday 30/06/2021 - 17:00 CEST

## Ged Corob Cook

### Counting irreducible modules for profinite groups

We say a profinite group G has UBERG if the number of irreducible G-modules of order k grows polynomially in k. This is equivalent to the completed group ring $\hat{\mathbb{Z}}[[G]]$ being generated with positive probability by n random elements, for some n (with respect to the Haar measure). I will talk about recent work, joint with S. Kionke and M. Vannacci, where we give algebraic conditions for G to have UBERG in terms of the sizes of the crown-based powers of monolithic primitive groups appearing as a quotient of G. As an application, we show that UBERG is not closed under extensions, unlike G being positively finitely generated (PFG). I will also discuss our work on a probabilistic version of the type FP1 condition, and some examples showing how these conditions relate to each other and to the PFG condition.

## Schedule January - April 2021

Wednesday 20/01/2021 - 17:00 CET

## Ariel Yadin

### Entropy realization on free groups and SL2(Z)

Abstract: We consider the following problem: Given a group G with a random walk on it, consider all possible ergodic invariant random subgroups (IRSs). Let J be the set of all random walk entropy values for these ergodic IRSs. It is immediate that J contains 0 and if the random walk is non-Liouville, it also contains the random entropy of the random walk on G. However, it is not clear what other values lie in J.

The phenomena of character rigidity seriously restricts the values of J, e.g. in the groups SLn(Z) for n at least 3. In light of this, it is interesting to understand what happens in SL2(Z) and in free groups.

We will show that J admits all possible values in these latter cases, which is the opposite phenomena than in the former higher rank groups. The solution to this problem is via a new type of IRS construction: intersectional IRSs.

All the notions will be explained, no prior knowledge is assumed.

Wednesday 03/02/2021 - 17:00 CET

## Ilir Snopche

### Test elements and retracts in free groups

Abstract: An element x of a group G is called a test element if for any endomorphism ϕ of G, ϕ(x) = x implies that ϕ is an automorphism. A subgroup R of a group G is said to be a retract of G if there is a homomorphism r : G → R that restricts to the identity on R. I will talk about test elements and retracts in free groups. In particular, I will discuss the following question raised by Bergman: Let F be a free group of finite rank and let R be a retract of F. Is it H ∩ R is a retract of H for every finitely generated subgroup H of F?

This talk is based on a joint work with Slobodan Tanushevski and Pavel Zalesskii.

Wednesday 17/02/2021 - 17:00 CET

## Aner Shalev

### Random Generation: from Groups to Algebras

Abstract: There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, with emphasis on finite simple groups. In this talk, based on a recent joint work with Damian Sercombe, we study similar notions for finite and profinite associative algebras.

Let A be a finite associative, unital algebra over a (finite) field k. Let P(A) be the probability that two random elements of A will generate A as a unital k-algebra. It is known that, if A is simple, then P(A) \to 1 as |A| \to \infty. We extend this result for larger classes of finite associative algebras. For A simple, we estimate the growth rate of P(A) and find the best possible lower bound for it. We also study the random generation of A by two special elements.

Finally, we let A be a profinite algebra over k. We show that A is positively finitely generated if and only if A has polynomial maximal subalgebra growth. Related quantitative results are also obtained.

Wednesday 24/02/2021 - 17:00 CET

## Sven Raum

### Why group theorists could care about operator algebras

Abstract: One of the foundational reasons to introduce operator algebras in the 1930's was the study of unitary representation theory, that is of a certain aspect of group theory. Ever since, group theory has provided important input and inspiration to operator algebraists. But what about group theorists? Why could they be interested in developments and questions from the field of operator algebras?

In this talk, I will illustrate my personal perspective on what the answer to this question could be. I will start by discussing selected historical examples of successful interaction between the fields, taking a birds perspective. Only then, I will rigorously introduce basic notations from operator algebras. A sample question on the relation between discrete groups and Polish groups comes forth from this discussion naturally. The final part of the talk will focus on recent development in C*-simplicity of discrete groups, which reveals new structure of groups and motivates questions in purely group theoretical terms.

Wednesday 10/03/2021 - 17:00 CET

## Gábor Pete

### Kazhdan groups have cost 1

Abstract: A probabilistic definition of groups with Kazhdan's property (T), due to Glasner & Weiss (1997), is that on any Cayley graph G of the group, for any ergodic group-invariant random black-and-white colouring of the vertices, with the density of each colour bounded away from 0, the density of edges connecting black to white vertices remains bounded away from zero. Amenable groups and free groups do not have property (T), while SL_d(\Z) with d\geq 3 do.

The cost of a transitive graph is one half of the infimum of the expected degree of invariant connected spanning subgraphs. Amenable transitive graphs and Cayley graphs of SL_d(\Z) with d\geq 3 have cost 1, while any Cayley graph of the free group on d generators has cost d, by Gaboriau (2000).

A question of Gaboriau aims to connect cost with the first L^2-Betti number of groups. For Kazhdan groups, the latter has been known to be 0 since Bekka & Valette (1997), and Gaboriau's question then suggests that the cost of any infinite Kazhdan Cayley graph should be 1. This is what we prove, in joint work with Tom Hutchcroft (Cambridge).

Wednesday 31/03/2021 - 17:00 CEST

## Gareth Wilkes

### Coherence of random groups

Abstract: Among the many properties one would wish a group to have is coherence: the property that every finitely generated subgroup is finitely presented. Among the 2-dimensional hyperbolic groups, which in some senses are 'generic' groups, coherence has been observed to have an empirical connection with Euler characteristic: those groups which are known to be coherent have nonpositive Euler characteristic. In this talk I will discuss joint work with Kielak & R. Kropholler which makes this connection probabilistic: a random group of negative Euler characteristic is coherent with high probability.

Wednesday 07/04/2021 - 17:00 CEST

## Alejandra Garrido

### Locally compact topological full groups

### Abstract: Topological (a.k.a piecewise) full groups of homeomorphisms of the Cantor set are a source of interesting examples of infinite simple groups. In the developing theory of totally disconnected locally compact (t.d.l.c.) groups, there is reason to look for examples that are simple and compactly generated. Piecewise full groups therefore seem an ideal place to look. Indeed, some well-known examples of compactly generated, simple, t.d.l.c. groups belong to this class, namely, Neretin's group of almost-automorphisms of a regular tree. I will report on joint work with Colin Reid and David Robertson on when and how piecewise full groups yield new examples of compactly generated, simple, t.d.l.c. groups.

Wednesday 14/04/2021 - 17:00 CEST

## Oren Becker

### Stability of approximate group actions

Abstract: An approximate unitary representation of a group G is a function f from G to U(n) such that f(gh) is close to f(g)f(h) for all g,h. Is every approximate unitary representation just a slight deformation of a unitary representation? The answer depends on G and on the norm on U(n). If G is amenable, the answer is positive for the operator norm on U(n) (Kazhdan '82). The answer remains positive if we use the normalized Hilbert-Schmidt norm and allow a slight change in the dimension n (Gowers-Hatami '15, De Chiffre-Ozawa-Thom '17). For both norms, the answer is negative if G is a nonabelian free group (or a nonelementary word-hyperbolic group). In this talk we shall discuss a similar notion where U(n) is replaced by Sym(n) with the normalized Hamming metric. We study the cases where G is either free, amenable or equal to SL_r(Z), r>=3. When G is finite, a slight variation of our main theorem provides an efficient probabilistic algorithm to determine whether a function f from G to Sym(n) is close to a homomorphism when |G| and n are both large. Based on a joint work with Michael Chapman.

## Schedule 2020

Wednesday 25/11/2020 - 17:30 CET

## Doron Puder

### Random permutations sampled by free words

Abstract: Fix a word w in a free group on r generators. A w-random permutation in the symmetric group S_N is obtained by sampling r independent uniformly random permutations \sigma_1,...,\sigma_r in S_N and evaluating w(\sigma_1,...,\sigma_r). Such w-random permutations have surprisingly rich structure with relations to deep results in geometric group theory. I'll survey some of this structure and state some conjectures.

This is based on joint works with Ori Parzanchevski and with Liam Hanany.

Wednesday 02/12/2020 - 17:00 CET

## Caterina Campagnolo

### Products of free groups in Lie groups

Abstract: For every Lie group G, we compute the maximal n such that an n-fold direct product of nonabelian free groups embeds into G. The proof is based on classification results for the corresponding Lie algebras.

This is joint work with Holger Kammeyer.

Wednesday 09/12/2020 - 17:00 CET

## Matthew Zaremsky

### Geometric embeddings into simple groups

Abstract: It is a classical fact that every finitely generated group embeds as a subgroup of a finitely generated simple group. In the 90's Bridson proved that if one relaxes "simple" to "no proper finite index subgroups" then such an embedding can be done in a quasi-isometric way. In joint work with Jim Belk, we prove that this is true even keeping the word "simple": every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. The simple groups we construct are "twisted" variants of Brin-Thompson groups. Certain of these twisted Brin-Thompson groups also provide examples of groups with interesting finiteness properties, and using them we can produce the second known family of simple groups with arbitrary finiteness properties (the first being due to Skipper-Witzel-Z).

Wednesday 16/12/2020 - 17:00 CET

## Benjamin Klopsch

### Strong Conciseness in Profinite Groups

Abstract: Based on joint work with Eloisa Detomi and Pavel Shumyatsky, I will discuss results and open questions regarding the (strong) conciseness of words in profinite groups. We will briefly explain the basic terminology and connect with the pre-history, recalling Philip Hall’s conjecture, positive results of Merzlyakov and others, and the counterexample of Ivanov. Then I will focus in on work of Detomi, Morigi and Shumyatsky that led a strengthened profinite version of Hall’s original conjecture. Finally, I plan to discuss the new evidence that we produced for this conjecture. In particular, every word is strongly concise in the class of nilpotent profinite groups, which leads to interesting applications and questions.