**Organizers**: Dominik Francoeur and Leo Margolis

This is a tentative list of the upcoming speakers for the Group Theory Seminar of the ICMAT. Click on the name of the speakers to see the title and the abstract.

Titles and abstracts for the talks can also be found at the Group Theory Seminar webpage on the ICMAT website.

**Time:** 11:30

**Place:** Aula Roja (IFT)

**Title:** Commuting probability for subgroups of a finite group

**Abstract:** If \(K\) is a subgroup of a finite group \(G\), the probability that an element of \(G\) commutes with an element of \(K\) is denoted by \(Pr(K,G)\). The probability that two randomly chosen elements of \(G\) commute is denoted by \(Pr(G)\). A well known theorem, due to P. M. Neumann, says that if \(G\) is a finite group such that \(Pr(G)\geq\epsilon>0\), then \(G\) has a normal subgroup \(T\) such that the index \( [G:T]\) and the order \(|[T,T]|\) are both \(\epsilon\)-bounded.

In the talk we will discuss a stronger version of Neumann's theorem: if \(K\) is a subgroup of \(G\) such that \(Pr(K,G)\geq\epsilon\), then there is a normal subgroup \(T\leq G\) and a subgroup \(B\leq K\) such that the indexes \( [G:T]\) and \( [K:B]\) and the order of the commutator subgroup \([T,B]\) are \(\epsilon\)-bounded.

This is a joint work with Eloisa Detomi (University of Padova).

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Hopfian wreath products and the stable finiteness conjecture

**Abstract:** *Wreath products* appear often as a case study in questions related to residual finiteness, thanks to a beautiful and simple characterization of Gruenberg. A related property is the Hopf property: a group is *Hopfian* if every self-epimorphism is an isomorphism. Every finitely generated residually finite group is Hopfian, which motivates looking at the Hopf property for wreath products, in hope of a simple characterization analogous to Gruenberg's.

It turns out that this problem is infinitely harder than Gruenberg's, even when focusing on the following special case: if G is finitely generated abelian, and H is finitely generated Hopfian, is \(G \wr H\) Hopfian? We will see that this question is equivalent to one of the most longstanding open problems in group theory: Kaplansky's *stable finiteness conjecture*, which is strongly related to the zero-divisor and idempotent conjectures, to the existence of a non-sofic group, and to Gottschalk's surjunctivity conjecture.

This is joint work with Henry Bradford (Cambridge).

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** TBA

**Abstract:** TBA

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** L^2-Betti numbers and algebraic fibring

**Abstract:** We introduce the theory of L^2-invariants and explain their relation to algebraic fibring of groups. In particular, if a group is virtually algebraically fibred, then its L^2-Betti numbers vanish. We also present a converse of this result for the class of virtually RFRS groups, which include RAAGs, RACGs, and special groups, as well as subgroups, products, and free products of all these groups. A positive characteristic version of this result is also presented.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Analogues of L2-Betti numbers in positive characteristic and homology growth

**Abstract:** For a group G satisfying the Atiyah conjecture, L2-Betti homology can be described as group homology with coefficients in the "Linnell-skew field" D(G), in which the group ring ℚG embeds. One approach to defining positive characteristic analogues of L2-Betti numbers is to find an analogous skew-field D_𝔽ₚG which contains the group ring 𝔽ₚG and take homology with these coefficients. In this talk, we will present some results about these positive characteristic invariants, in particular we will see how they relate to positive characteristic homology growth.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Maximal Subgroups of Thompson's group V

**Abstract:** Maximal subgroups of a group provide a range of information about the group. First, maximal subgroups correspond to primitive actions of a group. Secondly, in a finitely generated group every proper subgroup is contained in a maximal one. In this talk, we will discuss some ongoing work with Jim Belk, Collin Bleak, and Martyn Quick to understand and classify maximal subgroups of Thompson's group V.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Units in group rings and representation theory of groups of prime order

**Abstract:** The unit group \(U(\mathbb{Z}G)\) of the integral group ring \(\mathbb{Z}G\) of a finite group \(G\) is a long studied object, but nevertheless many fundamental questions remain open. The strongest possible expectation on finite subgroups of \(U(\mathbb{Z}G)\), expressed by Zassenhaus, had been that these always lie inside the trivial units \(\pm G\) up to conjugation in the bigger group algebra \(\mathbb{Q}G\). While this has been refuted in this most general form, it is still open for \(p\)-subgroups, and also whether the order of elements in \(U(\mathbb{Z}G)\) coincide with those in \(G\) remains unknown (after a suitable normalisation process).

I will explain a method which has been used to achieve several results on these questions, but the bottle neck of which, for the moment, is the understanding of representations of the most simplest groups one can imagine - the cyclic groups of prime order \(p\). Overcoming some of these difficulties, recent progress allows us to show that Zassenhaus' question has a positive answer at least for units of order \(p\), when the Sylow \(p\)-subgroup of \(G\) is also assumed to be of prime order.

This is joint work with Florian Eisele.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Group-graded algebras and the intrinsic fundamental group

**Abstract:** The intrinsic fundamental group of an algebra A is essentially the inverse limit of a diagram whose objects are groups which grade A in a connected way, and whose morphisms are group epimorphisms which correspond to quotient morphisms between these gradings. The computation of the intrinsic fundamental group consist of three steps:

1. Classify all the connected gradings of A up to equivalence of gradings.

2. Compute the quotient grading morphisms between the above grading classes.

These two steps yield a diagram of groups and homomorphisms.

3. To calculate the inverse limit of this diagram.

This is often the hardest of the above three challenges.

In this talk, we will focus on the intrinsic fundamental group of finite dimensional semisimple complex algebras and in particular matrix algebras and diagonal algebras. In these cases, the above diagram is always finite and the intrinsic fundamental group is its pullback.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Finite sets (containing zero) are mapping degree sets

**Abstract:** Let M,N be two oriented closed connected manifolds of dimension n. We define the mapping degree set as deg(M,N)={deg(f)| f:M-> N}. It is very relevant to construct inflexible manifolds M, i.e. deg(M,M) is bounded, and strongly inflexible manifolds M, i.e. for all N, deg(N,M) is bounded. They serve to produce functorial seminorms on n-manifolds.

On the other hand, one may ask which sets of integers can appear as deg(M,N) for some M,N. By cardinality reasons, not all sets can. Here we shall prove that any finite set of integers A, containing 0, is a mapping degree set for some choice. We extend this question to the rational homotopy theory setting, where an affirmative answer is also given, by using Sullivan models. (joint work with C.Costoya and A.Viruel)

**Time:** 11:30

**Place:** Aula 520, Módulo 17 (UAM)

**Title:** Acyclicity in bounded cohomology

**Abstract:** Bounded cohomology is an invariant of groups introduced by Johnson in the 70s and then extensively studied both in geometric group theory and in low dimensional topology after the pioneering paper by Gromov (1982).

Bounded cohomology is known to vanish in presence of amenable groups, but it is hard to compute for most groups (e.g. we do not know the full bounded cohomology of the non-abelian free group with two generators).

In this talk we will introduce bounded cohomology and discuss some recent computations involving boundedly acyclic groups (i.e. groups with trivial bounded cohomology in all positive degrees). We will report some results in collaboration with Francesco Fournier-Facio, Clara Löh and George Raptis.

**Time:** 10:00

**Place:** Aula Roja (IFT)

**Title:** Near actions

**Abstract:** I'll introduce the notion of near actions (of groups on sets). These are defined similarly as actions, except that each group element acts "outside a finite subset" and the action axioms are satisfied "outside a finite subset". I will present many examples, and notably discuss the realizability conditions: is a given near action induced by a genuine action?

**Time:** 11:30

**Place:** Aula Roja (IFT)

**Title:** The homology of big mapping class groups

**Abstract:** Big mapping class groups -- mapping class groups of infinite-type surfaces -- have recently become the subject of intensive study, having connections for example with geometric group theory and dynamical systems. However, their homology in degrees above one has so far been very little understood.

I will describe two contrasting results, from joint work with Xiaolei Wu, that exhibit very different behaviours of the homology of big mapping class groups. First, we find an uncountable family of big mapping class groups (including the mapping class group of the disc minus a Cantor set) whose integral homology vanishes in all positive degrees. Second, we find another uncountable family of big mapping class groups (including the mapping class groups of the flute surface and of the Loch Ness monster surface) whose integral homology is uncountable in each positive degree.

We also study the pure subgroups of big mapping class groups, namely the subgroups consisting of mapping classes that fix each end of the surface. These have more uniform behaviour: we prove that, for every infinite-type surface, its pure mapping class group has uncountable homology in each positive degree.

**Time:** 11:30

**Place:** Aula Roja (IFT)

**Title:** The Buzz About the BHZ: The Life and Times of Brauer’s Height Zero Conjecture

**Abstract:** In 1955, Richard Brauer, often regarded as the founder of modular representation theory, made one of the first of the so-called “local-global” conjectures in character theory and opened the door to an entire area of research. I’ll discuss the background of this conjecture, known as Brauer’s Height Zero Conjecture (BHZ), and its recent proof, which is joint work with G. Malle, G. Navarro, and P.H. Tiep.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** On generalisations of Thompson’s group V

**Abstract:**

In the 1960s R. Thompson defined three groups, F, T and V, all of which have shown to have surprising properties. For example, V was the first example finitely presented infinite simple group. In the early 2000s Nekrashevych and Matui realised these as topological full groups of some Cuntz-Algebras.

In this talk I will give an overview over some generalisations of V, especially those that are automorphism groups of Cantor-algebras. As it turns out, these can also be realised as topological full groups of certain groupoids coming from higher rank graphs. This is ongoing work with A. Vdovina.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Building trees out of compact subgroups

**Abstract:**

(Joint work with Pierre-Emmanuel Caprace and Timothée Marquis.) Compactly generated locally compact groups G have a well-behaved notion of ends, generalizing the number of ends of a finitely generated group: G has 0,1,2 or infinitely many ends, and having more than one end is associated to a certain kind of action on a tree (not necessarily of finite degree). It can also happen that the action on the tree is micro-supported, meaning that for each half-tree, there is an element fixing that half-tree pointwise but acting nontrivially on the opposite half-tree. The existence of micro-supported actions is in turn closely related to the structure of locally normal subgroups (closed subgroups with open normalizer) and has further implications for global properties of G, for instance it often leads to a nonamenable action of G on the Cantor space.

We find a sufficient condition in terms of a conjugacy class of compact subgroups for G to act on a tree in a way that shows G has infinitely many ends, and the action is also micro-supported. As an application, we obtain a connection between local and large-scale structure, for a class of groups acting on buildings that are obtained from Kac–Moody groups over finite fields. This is a class of totally disconnected locally compact groups where much is known about the group on a large scale (via the geometry of the building), but the structure of the profinite open subgroups is still mysterious.

**Time:** 12:00

**Place:** Aula Naranja (ICMAT)

**Title:** On Clifford theory

**Abstract:**

There are many open problems in ordinary and modular representation theory of finite groups and it is a general belief today that the way to approach these conjectures is
through a reduction of them to problems on finite simple groups, and then use the
Classification of Finite Simple Groups to solve them. By *reducing* we mean that our statement is true provided that some conditions are checked for every finite simple group.

To obtain these reduction theorems we need a powerful tool: Clifford theory. In this seminar, we explain this theory and we give an overview of some of the open problems in the area, making emphasis in the reduction theorems and the techniques involve to achieve them.

**Time:** 11:00

**Place:** Aula Gris 2 (ICMAT)

**Title:** Parabolic subgroups in even Artin groups of FC type

**Abstract:** We present a study of parabolic subgroups in even Artin groups through combinatorial and geometric means. The main result states that the collection of parabolic subgroups of a finitely generated even Artin group of FC type is closed under intersections; and as an application, we obtain a version of Tits alternative for the corresponding subgroups.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** The Atiyah conjecture

**Abstract:** The Atiyah conjecture states that the L²-Betti numbers of a countable group are always rational numbers, and a stronger version bounds the denominator of such numbers in terms of the torsion in the group. This conjecture has strong relations to other problems in group theory, such as the Lück approximation conjecture, Kaplansky's zero divisor conjecture, the strengthened Hanna Neumann conjecture and the problem of localization in group rings. In this talk we will explore how these questions and their pro-p variants are related, survey positive results among certain classes of groups and explain how these connections led to a recent solution of the strong Atiyah conjecture for pro-p free-by-cyclic groups (in joint work with Andrei Jaikin).

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Detecting free factors in profinite completions

**Abstract:**

If one is interested in finitely generated residually finite groups, it is natural to ask to what extent it is determined by its profinite completion; in other words, if two groups have isomorphic profinite completions, must they be isomorphic? This question is remarkably hard, even in the case where the profinite completion is a free profinite group.

A natural, perhaps more accessible, variant of the question is whether a free factor of a group G can be detected from its profinite completion: if G has a subgroup H, whose closure in the profinite completion of G is a profinite free factor, must H be a free factor of G? This question is still hard, with positive known answer only when G itself is a free group.

I will report on joint work with A. Jaikin in which it is shown that this question has positive answer when G is a virtually free group. The methods are different to those used previously and may be extended to other classes of groups if some challenging questions are answered on their completed group algebras.

I will attempt to keep the talk as accessible as possible, introducing the necessary background.

You can find more information and register on the conference website.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Oriented pro-p RAAGs and Galois cohomology

**Abstract:** One of the main open problems in recent Galois theory concerns the recognition of those profinite groups that occur as absolute Galois groups. Many attempts of detecting realizability of profinite groups as absolute Galois groups rely on cohomological aspects and many conjectures have been suggested. In particular, since a proof of the Bloch-Kato conjecture was provided in 2011, quadraticity of the group cohomology ring has become one of great interest. In a joint work with C. Quadrelli and Th. Weigel, we considered a certain class of groups generalizing the pro-p completion of classical RAAGs; our study corroborates the main conjectures in this context, and, on the other hand, it provides a huge class of pro-p groups that do not occur as absolute Galois groups. In fact, the class of the discrete right-angled Artin groups plays an important role in geometric group theory, since it generalizes both the classes of the free groups and of the free-abelian groups, and it provides a large test bed for those conjectures.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Verbal conciseness in profinite R-analytic groups

**Abstract:** A word w is said to be concise in a class C of groups if, for every group G in C such that w takes finitely many values in G, the verbal subgroup w(G) is finite.

There are few known classes of groups where all words are concise, and in this talk, we present a class of profinite groups, namely profinite R-analytic groups, with that property. These groups comprise an abstract group together with a manifold analytic structure over a general pro-p domain R in such a way that both structures are compatible.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Isomorphism Problem for Rational Group Algebras of Metacyclic Groups

**Abstract:** The Isomorphism Problem for group rings with coefficients in a ring R asks whether the isomorphism type of a group G is determined by its group ring RG. In general, it has a negative solution if no assumption is made about the ring or the group. For example, for abelian groups it has a positive solution if R is the field Q of rational numbers, but it has a negative solution in case R is the field of complex numbers. For metabelian groups it has a negative solution for every field, but a positive solution for R = Z, the ring of integers. With the aim to understand which property in between abelian and metabelian suffices for a positive solution in the case R = Q, we discuss the Isomorphism Problem for rational group rings of metacyclic groups. We prove a positive result under the assumption that G is nilpotent.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Aut-invariant quasimorphisms on groups

**Abstract:** For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. Joint work with Francesco Fournier-Facio.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Intersección de subgrupos parabólicos de grupos de Artin

**Abstract:** Los subgrupos parabólicos juegan un rol central en el estudio de los grupos de Artin. Motivados por lo que ocurre en grupos de Coxeter y grupos de trenzas estudiaremos la siguiente pregunta: ¿es la intersección de subgrupos parabólicos un subgrupo parabólico? Se conjetura que sí, y en esta charla responderemos afirmativamente esta pregunta cuando los grupos de Artin son dos-dimensionales y (2,2)-libres. Para ello introduciremos los complejos sistólicos-por-función y estudiaremos sus propiedades geométricas.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Some advances on the modular isomorphism problem

**Abstract:** Let R be a commutative ring and G a finite group. A fundamental problem in the study of the group algebra RG is to determine which features of G can be recovered from RG. An extreme formulation for this problem would be the question: Can the isomorphism type of G be recovered from RG? Although in this form it has negative answer in general, for some time it was conjectured that for a suitable choice of the ring R and/or imposing some extra properties on G, a positive answer could be obtained. One of these situations was the modular isomorphism problem, which asks whether one can recover the isomorphism type of G from kG, where G is a finite p-group and k is the field with p elements. We focus on this situation, and show some essential differences between the case p=2 (which is known to have a negative answer) and the case p>2 (which remains open). In order to do so, we find some natural relation between certain normal subgroubs of G and certain ideals of kG. We also use this relation to show that we can disregard abelian direct factors in the study of the modular isomorphism problem. The content is partially based in a joint work with Mima Stanojkovski and Ángel del Río.

**Time:** 11:00

**Place:** Aula Naranja (ICMAT)

**Title:** Some properties of the category of groups

**Abstract:** In a similar way that abelian categories appeared as a generalisation of abelian groups and modules, in 2002, semi-abelian categories arose as a categorical framework to capture groups and Lie algebras, among others [5]. In this talk we shall explore how the category of groups fits within this framework. It will be divided in two parts.

Firstly, we shall introduce a categorical characterisation of *Gp* inside of the category of monoids, i.e., a way to distinguish groups from monoids by only using universal properties [1, 6]. This can be generalised to broader settings considering groups in a monoidal category providing also interesting results in Hopf algebras.

In the second part, we shall recall several properties satisfied by the category of groups, focusing particularly on two of them: the existence of *algebraic exponents* and the *representability of actions*. Both of them are rather strong properties, in fact, they both have led to a categorical characterisation of the variety of Lie algebras amongst all varieties of non-associative algebras [2, 4]. These ideas together with the big resemblance between the categories of Groups and Lie algebras will allow us to state some open questions.

Joint work with Tim Van der Linden (Université catholique de Louvain, Belgium).

References

[1] X. García-Martínez, *A new characterisation of groups amongst monoids*, Appl. Categ. Structures 25 no. 4 (2017), 659–661.

[2] X. García-Martínez, M. Tsishyn, T. Van der Linden and C. Vienne *Algebras with representable representations*. Proc. Edinb. Math. Soc. 64 no. 3 (2021), 555–573.

[3] X. García-Martínez and T. Van der Linden, *A note on split extensions of bialgebras*, Forum Math. 30 no. 5 (2018), 1089–1095.

[4] X. García-Martínez and T. Van der Linden, *A characterisation of Lie algebras via algebraic exponentiation*, Adv. Math. 341 (2019), 92–117.

[5] G. Janelidze, L. Márki, W. Tholen, *Semi-abelian categories*, J. Pure Appl. Algebra 168 (2002), 367–386.

[6] A. Montoli, D. Rodelo, and T. Van der Linden, *Two characterisations of groups amongst monoids*, J. Pure Appl. Algebra 222 no. 4 (2018), 747–777.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** On some properties of groups generated by bireversible automata

**Abstract:** Automata are a powerful tool to construct groups with interesting and unusual properties. It seems natural to ask how the properties of an automaton are reflected in the properties of the group it generates. In this talk, we will be interested in a special class of automata known as bireversible. Groups generated by such automata remain elusive, and very few examples are known. We will investigate some of their properties, thus yielding restrictions on the groups that can be generated by such automata. Notably, we will obtain information of the distortion of cyclic subgroups of these groups.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Computations of Bredon homology of Artin groups of Dihedral type.

**Abstract:** An Artin group of Dihedral type is a group with a presentation consisting on two generators, x and y and a single relation of the form (xy)^n = (yx)^n or y(xy)^n = x (yx)^n. We will make some computations of the Bredon Homology of these groups with respect to the family of virtually cyclic subgroups. We then will use this computations to obtain a information about K-theory groups of the group ring of a dihedral Artin group using the Farrell-Jones isomorphism. This is a joint work with Ramón Flores.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Detecting blocks from the character table

**Abstract:** Many properties of a finite group $G$ can be detected from its complex character table $X(G)$. We are particularly interested in local properties, i.e. data related to a Sylow $p$-subgroup $P$ of $G$. It has been shown that $X(G)$ determines whether $P$ is abelian and in that case the isomorphism type of $P$ is determined as well. In this work we replace $P$ by a defect group of a block of $G$ and show (among other things) that the exponent of the center $\exp(\Z(D))$ is determined by $X(G)$.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Non convergence of eigenvalue measures associated to residual chains

**Abstract:**

Let \(G\) be a discrete residually finite group and let \(A \in \text{Mat}_n(\mathbb{C}[G]). \) Let \(G \unrhd N_1 \unrhd N_2 \ldots\) be a chain of normal subgroups of finite index with trivial intersection. Set \(G_i = G / N_i. \)Then \(A\) acts by right multiplication via reduction modulo \(N_i\) on \(\mathbb{C}[G_i]^n\). Since \(\mathbb{C}[G_i] \cong \mathbb{C}^{\vert G_i \vert}\) as \(\mathbb{C}\)-vector spaces, this action can be represented by a matrix \(A_i \in \text{Mat}_{n \cdot \vert G_i \vert}(\mathbb{C}).\) For every \(i\) let now \(\lambda_1^{(i)}, \ldots , \lambda_{n \cdot \vert G_i \vert }^{(i)} \) be the eigenvalues of \(A_i\) and define

\(\mu_i = \frac{1}{\vert G_i \vert}\sum\limits_{k = 1}^{n \cdot \vert G_i \vert } \delta_{\lambda_k^{(i)}},\)

Where \(\delta_c\) denotes the Dirac measure at \(c \in \mathbb{C}.\)

We now can ask the following questions:

(1) Does the limit \(\lim\limits_{i \to \infty} \mu_i (\{0 \})\) exist?

(2) If the answer to the first question is yes, does the limit depend on the chain \(\{N_i\}\)?

(3) Let \(\mathcal{N} (G) \) denote the group von Neumann algebra. We can consider \(A\) as an element of the tracial von Neumann algebra \(\text{Mat}_n(\mathcal{N} (G)) \) acting on the Hilbert space \( (\ell^2 (G))^n. \) Therefore we can define the Brown measure \(\mu_A\) of \(A\) and ask: Does the sequence \(\mu_i\) converge weakly to \(\mu_A\)?

All these questions have been already studied for the case when \(A\) is normal. In this case the answer to all questions is positive. In this talk we will show that in general the answers to questions (2) and (3) are negative. This is a joint work with Andrei Jaikin.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** On the Virtual Structure Problem for group rings

**Abstract:** In representation theory, over a domain R, a finite group G is studied through the module category of the group ring RG. The emphasis of this talk will be the role of infinite group theory in this. More concretely, in the first part we will recall some of the motivating conjectures on group rings (such as the Zassenhaus conjectures and the Isomorphism problem) and also the virtual structure problem. Thereafter we will present a kind of program to the latter that is behind recent trends and results in the field (which is a mixture of representation theory and (geometric) group theory). This will be illustrated through ongoing work with Doryan Temmerman where cases of the aforementioned conjectures are studied through the use of amalgamated products.

**Time:** 11:30

**Place:** Aula Gris 2 (ICMAT)

**Title:** Which amalgamated products exist in \(GL_n(D)\)?

**Abstract:** Given two finite subgroups \(G, H \in GL_n(D) \), with \(D\) a \(\mathbb{Q}\)-division algebra, we will discuss the existence of \(G \star_{G \cap H} H\) inside \(GL_n(D) \). The focus of the main part of the talk will be on explaining explicit constructions. In the last part we will turn the attention to the quantity of so called ping-pong partners for a given finite subgroup. More precisely it will turn out that inside \(GL_n(\mathcal{O})\), for \(\mathcal{O}\) an order in \(D\), they form a profinite dense set. This talk is based on joint (ongoing) work with Doryan Temmerman.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Representations over finite fields, Probability and Zeta Functions

**Abstract:** In this talk we will study representation growth of a profinite group G over finite fields. On one hand, having "asymptotically few" representations over finite fields connects to (surprising) probabilistic generation properties of the completed group-ring of G. On the other hand, we can encode the number of representations of G over finite fields into a zeta-function and we will investigate how analytic properties of this zeta-function connect to properties of the original group G and its completed group-ring. This is joint work with Ged Corob-Cook and Steffen Kionke.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Relative order and spectrum: three questions and two answers

**Abstract:** We consider a natural generalization of the concept of order of an element in a group: an element g �?G is said to have (relative) order k in a subgroup H of G if k is the first strictly positive integer such that g^k �?H. We use Stallings-like automata to study this notion and its algorithmic properties in the realm of free groups and some related families. In this talk, I will briefly survey the main concepts introduced in this work, discuss some natural questions on them, and present both positive and negative (algorithmic) results we have obtained. This is joint work with Enric Ventura and Alexander Zakharov.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Asymptotically rigid mapping class groups

**Abstract:** During this talk we will focus on a family of asymptotically rigid mapping class groups which includes a braided version of Ptolemy-Thompson groups. We will build a contractible cube complex on which these groups act and we will see which kind of properties can be deduced from this action.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Portraits and patterns in contracting groups

**Abstract:** Any automorphism of a \(d\)-regular rooted tree \(T\) can be described by decorating the tree vertices with permutations from the symmetric group \(S_d\). If one cuts this decoration at a certain level of the tree, we obtain a finite pattern.
On the other hand, if the action of a group \(G\) on \(T\) is what is known as contracting, each group element can be described by a finite decoration made up of a pattern on the top levels and group elements from a finite set on the leaves.
Given a contracting group, a natural question to ask is which are the patterns and portraits seen on the group. We point out some relations between these two notions, look at their growth for regular branch groups and analyse the situation in some particular examples.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** What we know about Donovan’s conjecture

**Abstract:** One of the major open questions in the representation theory of finite groups over fields of characteristic p>0 is Donovan’s conjecture, which constrains the representation theory of a group G and the structure of its group algebra in terms of the isomorphism type of its Sylow p-subgroups. I will talk about some recent results by myself and others on this conjecture and related questions.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Characters and generation of Sylow subgroups

**Abstract:** In this talk I will present a characterization of the groups possessing 2-generated Sylow 2-subgroups in terms of their character values (the content is based on joint works with G. Navarro, N. Rizo and A. A. Schaeffer Fry).

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Simplicity of Nekrashevych algebras of contracting self-similar groups

**Abstract:** A self-similar group is a group \(G\) acting on a regular, infinite 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 that 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 result of the speaker and Benjamin Steinberg 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 GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** On the class size prime graph of finite groups

**Abstract:** Let \(G\) be a finite group, and \(cs(G)\) be the set of its conjugacy class sizes. Many researchers in the literature have shown the in-depth relationship that exists between the arithmetical properties of \(cs(G)\) and the structure of \(G\). In the last decades, the use of the *class size prime graph* is gaining an increasing interest. This (simple undirected) graph has the prime divisors of the numbers in \(cs(G)\) as set of vertices, and the edges are pairs \(\{p, q\}\) such that \(pq\) divides some number in \(cs(G)\).

The aim of this talk is to present some current results, obtained in joint work with S. Dolfi, E. Pacifici and L. Sanus, about this subject.

See the website for more information.

See the website for more information.

**Time:** 11:30

**Place:** Aula Naranja (ICMAT)

**Title:** Can we put arrows in RAAGs?

**Abstract:** In the realm of Geometric Group Theory, the family of right-angled Artin groups has gained more and more importance through the years. They generalize at the same time both free and free abelian groups, and moreover their combinatorial nature has led to proving important results. On top of that, they also have a rich geometric nature. But what happens if we put some arrows in the defining graph of a RAAG? During this seminar we will discuss the consequences of such decision and we will see the analogies and the differences between classical and oriented RAAGs. If time permits, we will also talk about some results related to their pro-2 completion.

**Time:** 11:30

**Place:** Aula Gris 2 (ICMAT)

**Title:** Small covers of big surfaces

**Abstract:** Covering spaces are ubiquitous in topology. This talk will be about finite-sheeted covers of surfaces. In particular, one can ask; given two surfaces, when does one admit such a cover over the other.
A result of Massey provides the answer when the surfaces are finite-type. In joint work with Ty Ghaswala, we look at what can be said about the remaining (uncountably many) cases.