Madrid Group Theory Seminar

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Wednesday(!) 29th May.  11:30.  ICMAT, Aula Gris 1

Oihana Garayalde (Universidad del País Vasco)

Cohomology and Carlson's depth conjecture

Providing the cohomology ring of a finite group can be intrinsically hard. Instead, it is desirable, and sometimes satisfactory, to describe certain ring invariants in terms of group theoretic properties. Thanks to Quillen's Stratification Theorem, the Krull dimension of a cohomology ring of a finite group coincides with the p-rank of the given group. Here, p denotes a prime number, and the p-rank is the maximal rank of the elementary abelian p-subgroups.

Nevertheless, the depth seems to be a more intricate ring invariant. Although there are some known upper and lower bounds for this value, none of them seem to be easy to compute. In this talk, we present a conjecture of J. F. Carlson that deals with the depth of cohomology rings, and present an infinite family of finite p-groups that satisfy the aforementioned conjecture.

Tuesday 21st May.  11:30.  ICMAT, Aula Gris 2

Davide Perego (Universidad de Sevilla)

Context-free graphs and their transition groups

 Chomsky hierarchy for groups is a (still incomplete) way to classify groups via their word problem languages. After a brief introduction on the state of the art, we will define a new family of co-context-free groups and discuss the connections between this and other interesting examples and families.

Tuesday 7th May.  10:00 (!).  ICMAT, Aula Gris 2

Caterina Campagnolo (UAM/ICMAT)

El grupo genérico numerable es acotadamente acíclico

En esta charla quisiera enfocar en una aplicación sorprendente de un resultado reciente obtenido con Fournier-Facio, Lodha y Moraschini: la coomología acotada del grupo genérico numerable es cero en cada grado positivo. Esto contrasta fuertemente con el comportamiento de los grupos genéricos obtenidos por el modelo de Gromov, que son iperbólicos y por tanto tienen coomología acotada altamente no trivial.

Empezaré dando contexto y motivación para el estudio de la coomología acotada.

Tuesday 30th April.  11:30.  ICMAT, Sala Naranja

Sangrok Oh (Universidad del Pais Vasco)

Large scale geometry of graph 2-braid groups

The intersection complex of the universal cover of a special square complex is a quasi-isometry invariant which encodes the information that quasi-flats are preserved by a quasi-isometry up to finite Hausdorff distance.

In this talk, we will use the intersection complex to classify a certain class of graph 2-braid groups, which are the fundamental groups of special square complexes, up to quasi-isometries.

Moreover, we will see when such a graph 2-braid group is quasi-isometric to a right-angled Artin group. 

This is a joint work with Byunghee An.

Tuesday 23th April.  11:30.  ICMAT, Sala Gris 2

Eduard Schesler (University of Hagen)

Just-infinite groups via iterated semidirect products

An infinite group G is called just-infinite if all of its proper quotients are finite. Since its introduction by McCarthy in the late 1960's, the class of just-infinite groups received a lot of attention. One reason for the importance of just-infinite groups is that, by Zorn's lemma, every finitely generated, infinite group admits a just-infinite quotient. By a celebrated result of Wilson, the study of just-infinite groups can be reduced to the study of simple groups, branch groups, and hereditarily just-infinite groups, i.e. groups all of whose finite index subgroups are just-infinite. After recalling some background on residually finite groups and profinite completions, I will present an elementary idea that gives rise to constructions of all 3 types of just-infinite groups. As an application, we will discuss the first examples of finitely generated just-infinite groups that are residually finite and have positive rank gradient. In fact we will see that these examples have positive first L2-Betti-number. This talk is based on joint work with Steffen Kionke.

Thursday 4th April.  11:30.  ICMAT, Sala Naranja

Macarena Arenas (University of Cambridge)

Cohen-Lyndon-type properties and asphericity in two and more dimensions

In its classical form, the Cohen-Lyndon property encodes independence between the relators in a group presentation. It is an interesting structural property that has been proven to hold, in one form or another, for various classes of groups. In this talk I will tell you a little bit about how this property arises naturally in connection with asphericity, and I will discuss some examples.

Tuesday 19th March.  11:30.  ICMAT, Sala Naranja

Iván Chércoles (UCM/ICMAT)

The space of left-preorders of a free product

The notion of left-order on a group has been studied before by many authors, as it is a useful tool to study infinite groups. We can define a topology on the set of left-orders of a given group. This topological space is called the space of left-orders. It is known that the space of left-orders of a finitely generated free product is a Cantor set. In this talk, we will consider left-preorders, a recent generalization of the notion of left-order. We will define the space of left-preorders and we will discuss some of its properties. We will conclude that the space of left-preorders of a finitely generated free product is a Cantor set.

Tuesday 12th March.  11:30.  ICMAT, Sala Naranja

Juan Martínez Madrid (Universidad de Valencia)

Covering the set of p-elements of a finite group

There exists many papers studying the number of proper subgroups needed to cover a finite group. In this talk, we are interested in slightly different questions. How many proper subgroups do we need to cover the set of p-elements of a group? Do we need all Sylow $p$-subgroups to cover the set of p-elements of a group?

Tuesday 5th March.  11:30.  ICMAT, Sala Naranja

Rögnvaldur G. Möller (University of Iceland)

Graphs of group actions

This is a preliminary report on work in progress. Group actions on trees, in various forms, play a central role geometric group theory and the theory ot totally disconnected, locally compact groups.  What I want to describe in this talk is a method to construct group actions on trees.  The concepts of a graph of group actions and its fundamental group are inspired by Bass-Serre theory and the Burger-Mozes construction. The definition of a graph of group actions resembles the definition of a graph of groups in Bass-Serre theory, except that in stead of vertex and edge groups and embeddings of groups we have group actions and embeddings of group actions.  The fundamental group of a graph of group actions is then constructed using ideas similar to those used by Burger and Mozes in the construction of their universal groups.  Using this construction we recover the fundamental group of a graph of groups in Bass-Serre theory and the Burger-Mozes universal groups. It is possible to construct groups acting on trees with various conditions on the "local" action, e.g. groups acting on a tree with a prescribed action on balls of radius k, extending previous results of Tornier.

Joint work with Florian Lehner (University of Auckland), Christian Lindorfer (TU Graz) and Wolfgang Woess (TU Graz).

Tuesday 27th February.  11:30.  ICMAT, Sala Naranja

Pavel Zalesski (University of Brasilia)

Prosoluble subgroups of free profinite products

We shall discuss prosoluble subgroups of free profinite products towards a complete characterization of them in terms of the intersections with the free factors.

Tuesday 20th February.  11:30.  ICMAT, Sala Naranja

Eduardo Tablate (ICMAT)

The local geometry of idempotent Schur multipliers

Schur multipliers are linear maps defined on matrix algebras with considerable influence across geometric group theory, operator algebras and functional analysis. Haagerup's groundbreaking investigations into free groups and subsequent inquiries into semisimple lattices paved the way for understanding the deep geometric properties encoded through approximation properties of Schur multipliers defined on their matrix algebras. More recently, stronger advances in the study of high-rank lattices have uncovered stronger rigidity properties, particularly through the exploration of Lp-approximations. This exploration reveals intriguing pathologies, such as the absence of Lp-approximations via Fourier or Schur multipliers over SLn(R), leading to a strong form of nonamenability with possible implications in the classification of certain von Neumann algebras. In this talk, we will consider idempotent Schur multipliers, that is the ones defined by characteristic functions. Our main result extends the well-known Feffermann ball multiplier theorem, revealing deep parallels between Schur and Fourier multipliers. As an application we fully characterize the local Lp-boundedness of smooth idempotent Fourier multipliers on connected Lie groups, completing, for Lie groups, the search of Fourier Lp-idempotents. This is joint work with Javier Parcet and Mikael de la Salle.

Tuesday 13th February.  11:30.  ICMAT, Sala Naranja

Oussama Hamza (Western University)

On extensions of number fields with given quadratic algebras and cohomology

At the beginning of the century, Labute and Minac introduced a criterion, on presentations of pro-p groups, ensuring that the cohomological dimension is two. Groups with presentations satisfying this condition are called mild. In this talk, we introduce a new criterion on the presentation of finitely presented pro-p groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two.  We interpret these groups as Galois groups over p-rational fields with prescribed ramification and splitting.

Tuesday 6th February.  11:30.  ICMAT, Sala Naranja

Antonio Viruel (Universidad de Málaga)

Path partial groups

It is well known that not every finite group arises as the full automorphism group of some group. In this lecture we show that the situation is dramatically different when considering the category  of partial groups, as defined by Chermak: given any group H there exists infinitely many non isomorphic partial groups M such that Aut(M)=H. To prove this result, given any simple undirected graph G  we construct a partial group P(G), called the path partial group  associated  to G, such that Aut(P(G))=Aut(G).

This is a joint work with Antonio Díaz Ramos (Málaga) and Rémi Molinier (Grenoble).

Tuesday 30th January.  11:30.  ICMAT, Sala Gris 2

Andrei Jaikin (UAM/ICMAT)

Coherence of one-relator groups and their group algebras

In my talk, I will explain the main ideas of how to prove that one-relator groups and their group algebras over fields of characteristic zero are coherent. This solves a well-known problem of Baumslag. These results are based on joint work with Marco Linton.

Tuesday 12th December.  11:30.  ICMAT, Sala Naranja

Henrique Souza (UAM/ICMAT)

Rational representations and Lück approximation

How singular is an element in the group algebra? Given an element x inside the complex group algebra of a semisimple linear algebraic group G, we raise a conjecture about the asymptotic behaviour of the dimensions of the kernels ker(x) under the various irreducible rational representations of G. Explicitly, it is expected that this dimensions approximate the dimension of the kernel of x as an operator inside a von Neumann algebra associated to G. We confirm this conjecture when G = SL(2,C) using profinite methods, and as an application we derive a new method of computing the L²-Betti numbers of hyperbolic 3-manifolds. (Joint work with Andrei Jaikin and Lander Guerrero Sánchez)

Tuesday 5th December.  11:30.  ICMAT, Sala Naranja

Rodrigo de Pool (ICMAT)

Braided multitwists in the mapping class group

The study of Dehn twists and their relations has shown to be fruitful in the understanding of mapping class groups. In this talk, we will describe which  products of commuting Dehn twists  satisfy the braid relation. Time permitting, some applications will be discussed.

Tuesday 28th November.  11:30.  ICMAT, Sala Naranja

Pablo Sanchez-Peralta (UAM/ICMAT)

On vanishing criteria of L^2-Betti numbers of groups

The vanishing of the L^2-Betti numbers of a countable discrete group have proved to be a powerful tool to detect structural properties of the group. The aim of this talk will be to show how the L^2-Betti numbers of subgroups satisfying certain normality conditions produce the vanishing of the L^2-Betti numbers of the whole group. Additionally, we shall exhibit an algebraic proof of a celebrated theorem of Gaboriau, addressing a request of Bourdon, Martin and Valette.

Tuesday 21st November.  11:30.  ICMAT, Sala Naranja

Dario Ascari (University of the Basque Country)

A finitely presented group with a cohomology class which is weakly bounded but not bounded

The subtle distinction between the notion of bounded cohomology class and weakly bounded cohomology class plays a role in several applications. However, it's not easy to construct groups where the two notions aren't equivalent. We provide the first known example of such a finitely presented group. In particular, this allows us to answer to a question of Gromov about De Rham cohomology of closed manifolds.

Tuesday 14th November.  11:30.  ICMAT, Sala Gris 2

Marco Linton (University of Oxford)

Virtually free-by-cyclic groups, one-relator groups and coherence

The aim of this talk will be to show that under certain geometric conditions, virtually free-by-cyclic groups can be characterised in terms of homological invariants. Using this characterisation, I will then explain how many groups of cohomological dimension two that are known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, I will show that one-relator groups with torsion are virtually free-by-cyclic, resolving a conjecture of Baumslag's. (Joint work with Dawid Kielak).

Tuesday 7th November.  11:30.  ICMAT, Sala Naranja

Cyril Lecuire (ENS Lyon)

Fundamental groups of 3-manifolds

 I will introduce some properties of fundamental groups of 3-manifolds and explain their relations with the topology and geometry of the corresponding 3-manifolds.

Tuesday 31st October.  11:30.  ICMAT, Sala Naranja

Paloma López Larios (UCM)

Counting subgroups using Stallings automata and generalizations

The problem of counting finite index subgroups of the free group was tackled in 1949 by Marshall Hall, who provided a recursive formula for the number of subgroups of a given finite index in a free group of finite rank. In this talk, we will review some main ideas of Stallings automata theory and we will apply them to prove Hall’s result. Moreover, we will see how to obtain a similar formula in the case of free times free-abelian groups, for which we will use enriched automata, a generalization of Stallings automata. This work was developed in my Master’s thesis under the supervision of Jordi Delgado.

Tuesday 17th October.  11:30.  ICMAT, Sala Naranja

Richard Mandel (University of the Basque Country)

The quadratic Diophantine problem in Baumslag-Solitar groups

The Diophantine problem for a group G is the problem of deciding whether a given equation has a solution in G. The restriction of this problem to the class of quadratic equations (where each variable appears twice) is an important variation which has been extensively studied in various classes of groups (free, hyperbolic, free metabelian etc.). In this talk, I will discuss some decidability and complexity results for the quadratic Diophantine problem over the Baumslag-Solitar groups, with an emphasis on the groups BS(1,n) and BS(n,\pm n) (based on joint work with Alexander Ushakov).

Tuesday 3rd October.  11:30.  ICMAT, Sala Naranja

Yago Antolin (UCM / Heriot-Watt University)

The Strongest Tits Alternative

The strongest Tits alternative is the following dichotomy for a group G: every subgroup of G is either abelian or maps onto a non-abelian free group. We will review some facts about this alternative, and we will sketch a proof of the fact that even Artin groups of FC-type virtually satisfy this alternative. This is a joint work with Islam Foniqi.

Talks 2022-23 

Talks 2016-22