Regular research seminar in Group Theory
Organiser: Andrei Jaikin (ICMAT) and Leo Margolis (UAM/ICMAT)The next talk
10 December at 16:30, ICMAT, Sala Naranja
Joaquin Brum (Universidad de la República Uruguay), Moduli spaces for group actions on the line.
In this talk, I will discuss the moduli space of actions of a finitely generated group G on the real line. I will first explain what we mean by this space, and then describe its structure in the cases of solvable and Thompson-like groups. Finally, if time permits, I will present some open problems concerning the structure of these spaces for amenable groups. This is a joint work with Nicolás Matte Bon, Michele Triestino and Cristóbal Rivas.
Talks 2025-2026
15 December at 11:30, ICMAT, Gris 2
Michael Kopreski (Universidad del País Vasco), Geometric models for infinite-type surface mapping class groups
Let S be an infinite-type surface and let G < Map(S) be a locally bounded Polish subgroup. We construct a metric graph M of simple arcs and curves on S preserved by the action of G and for which the vertex orbit map is a coarse equivalence; moreover, if G is boundedly generated, then the orbit map is a quasi-isometry. We show that if S contains a non-displaceable subsurface and G > PMap_c(S) is boundedly generated or G = Map(S) and is locally bounded, then asdim M = asdim G is infinite.
10 December at 16:30, ICMAT, Sala Naranja
Joaquin Brun (Universidad de la República Uruguay), Moduli spaces for group actions on the line.
In this talk, I will discuss the moduli space of actions of a finitely generated group G on the real line. I will first explain what we mean by this space, and then describe its structure in the cases of solvable and Thompson-like groups. Finally, if time permits, I will present some open problems concerning the structure of these spaces for amenable groups. This is a joint work with Nicolás Matte Bon, Michele Triestino and Cristóbal Rivas.
3 December at 16:30, ICMAT, Sala Naranja
Florian Eisele (University of Manchester), Block theory of profinite groups
Profinite groups are a class of topological groups with close ties to finite groups. Many of the questions around group rings of finite groups and their representation theory have profinite analogues. I will report on results that generalise block fusion systems and Puig's theory of nilpotent blocks to the profinite setting, and a resulting description of blocks with infinite dihedral defect groups. This is joint work with MacQuarrie and Franquiz Flores.
26 November at 16:30, ICMAT, Sala Naranja
Karol Duda (Universidad del País Vasco), Locally elliptic actions on small cancellation complexes.
Classical and graphical complexes satisfying small cancellation conditions are
one of examples of nonpositively curved spaces. It is conjectured that every locally
elliptic action of a finitely generated group on a finite-dimensional nonpositively curved complex is elliptic. We show some results concerning this conjecture in the cases of classical and graphical small cancellation complexes. This is based on joint works with Martin Blufstein and Huaitao Gui.
19 November at 16:00, ICMAT, Sala Naranja
Aram Dermenjian (Universidad de Sevilla), The strong exchange property for Coxeter matroids
This talk assumes no prior knowledge for any of the terms mentioned in this abstract. Matroids are a combinatorial structure that abstracts linear independence. They have a deep connection with Grassmannians, the variety of subspaces of a vector space cut out by the Plücker equations. Grassmannians and matroids are inherently 'type A' objects and have generalizations to other root systems giving rise to what is known as Coxeter matroids. As such, one can generalize (through representation theory) these Grassmannians and Plücker equations to other Coxeter types using miniscule varieties and equations that define the quotient G/P of a simply connected complex Lie Group G by a maximal parabolic subgroup P with a miniscule fundamental representation. In addition, Borovik, Gelfand and White give a description of a strong exchange property for Coxeter matroids. It turns out there is a strong link between the strong exchange property and the tropicalization of the equations that define G/P. In this talk, we describe this link through representation theory, tropical geometry and polytope theory. (This is joint work with Kieran Calvert, Alex Fink and Ben Smith).
12 November at 16:30, ICMAT, Sala Naranja
Gabriel A. L. Souza (Universidad de Valencia), Explicit free groups in division rings
In 1977, A. Lichtman formally posed the question on whether the multiplicative group
of a division ring always contains a nonabelian free group. This conjecture has been widely studied, and many important cases have been settled. In this talk, we explore results that allow us to explicitly construct elements generating nonabelian free groups in some of the main classes of division rings, such as quaternion algebras and Malcev-Neumann series rings. This is joint work with Professor Jairo Z. Gonçalves.
5 November at 16:30, ICMAT, Aula Gris 3
Owen Garnier (Universidad de Sevilla), Normalizers of parabolic subgroups in Artin and Coxeter groups
Parabolic subgroups are important subgroups defined in Coxeter groups and their associated Artin groups. In both settings, the normalizer of a parabolic subgroup decomposes as a semidirect product which is understood combinatorially by introducing a suitable groupoid. For Coxeter groups, this groupoid is the Brink-Howlett groupoid, for Artin groups it is the ribbon groupoids.
This talk will start by presenting these "classical" groupoids. I will then present a joint work with E. Heng, A. Licata and O. Yacobi, in which we provide a geometric interpretation of these groupoids using the so-called Tits cone intersection attached to a parabolic subgroup.
29 October at 16:00, ICMAT, Aula Naranja
Simone Virili (UAB), A study of Sylvester rank functions via functor categories.
Given a ring , a Sylvester rank function on the category of finitely presented right -modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value on .
In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left -modules to Abelian groups. This enlargement of the setting comes with many advantages and, to illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.
22 October at 16:30, ICMAT, Aula Naranja
Letizia Issini (Université de Genève), Divergence in groups with microsupported actions
The divergence of a group is a quasi-isometry invariant that measures how difficult it is to connect two points avoiding a large ball around the identity. It is easy to see that it is linear for direct products of infinite groups, and from that to deduce that it is linear for branch groups. In this talk, I will discuss divergence for weakly branch groups and, more in general, groups with a micro-supported action by homeomorphisms on a topological space. This is ongoing joint work with D. Francoeur and T. Nagnibeda.
15 October at 16:30, ICMAT, Aula Naranja
Nansen Petrosyan (University of Southampton), Group theoretic Dehn fillings and their L^2-Betti numbers
Dehn filling is a fundamental tool in group theory, appearing in the solution of the Virtual Haken Conjecture, the study of the Farrell-Jones Conjecture, the isomorphism problem of relatively hyperbolic groups, and the construction of purely pseudo-Anosov normal subgroups of mapping class groups. In this talk, I will discuss past joint work with Bin Sun on the cohomology of Dehn filling quotients and our recent results on their L^2-Betti numbers. The applications include the verification of the Singer Conjecture for certain Einstein manifolds, virtual fibering, and the construction of new examples of hyperbolic groups with exotic subgroups.
8 October at 16:30, ICMAT, Aula Naranja
Gonzalo Ruiz (ICMAT), Complex hyperbolic representations of groups: the infinite-dimensional setting
Part of the content is a joint effort with Nicolas Monod. We will introduce tools that allow us to study, deform, and define invariants for the isometric representations of groups into complex hyperbolic spaces. These tools were constructed by drawing an analogy with the real negative-type functions as well as with previous constructions by Py-Monod and Monod.
We will describe some results concerning the representations of the groups PU(1,n) and PO(1,n) that can be obtained with these tools.
1 October at 16:30. ICMAT, Aula Naranja.
Sam Corson (UPM), Many trivially generated groups are the same
When the algebraic generators of a group are all trivial then the group is also trivial. However, when the group is combinatorially described using infinite words, the "generators" can be trivial while the group is uncountable. I will give background and present work showing that many natural examples of such groups are isomorphic.
Talks 2024-2025
La regularidad crítica para una acción \Phi de un grupo en una variedad diferenciable intenta detectar qué tan suave puede hacerse \Phi mediante conjugaciones topológicas. En esta charla motivaremos y daremos una definición formal de lo anterior, y revisaremos resultados en el contexto de acciones de grupos en el intervalo o el círculo. Pondremos énfasis en acciones de grupos nilpotentes y solubles.
In this talk I will introduce a deformation space for connected large type Artin groups. Under mild hypotheses, the space depends only on the isomorphism type of the Artin group (not the presentation graph), and in these cases admits an action of the outer automorphism group. I will discuss how this action can be used to show the outer automorphism groups of such Artin groups are of Type VF (which in particular implies finitely presentability).
My thesis explores how concepts of formal language theory can be used to study left-orderable groups. It analyses the languages formed by their positive cones and demonstrates how the abstract families of languages (AFLs) in the Chomsky hierarchy (in particular regular and context-free languages) interact with core group-theoretic constructions under subgroups, extensions, finite generation and taking direct products with the integers.
In this pre-defence, I will present the main thesis results, explain how they relate to the state of the art, and provide proof sketches to highlight the main ideas to the audience.
Braid groups are examples of both finite-type Artin groups and mapping class groups. One important tool in the study of mapping class groups is the marking graph, a graph whose vertex set consists of certain collections of curves on a surface S and which is quasi-isometric to the mapping class group of S. In this talk, I will recall Masur and Minsky’s definition of the marking graph for surfaces and I will define an analogue of the marking graph which is quasi-isometric to a finite-type Artin group modulo its center.
Profinite rigidity of groups asks whether the isomorphism type of a group can be determined entirely in the collection of its finite quotients. In this talk, I will introduce the study of profinite rigidity in a different category, namely the category of modules over a Noetherian domain Λ. I will present some foundational results and explore which properties of Λ-modules are profinitely invariant. I will explain how these properties may be used to obtain profinite rigidity results for free Λ-modules under a homological assumption on Λ and for all modules in the case where Λ is a Dedekind domain or PID. Finally, I will sketch some applications of this theory for the profinite rigidity of certain classes of solvable groups.
A cobordism W between compact manifolds M and M’ is an h-cobordism if the inclusions of M and M’ into W are both homotopy equivalences. This kind of cobordisms plays an important role in the classification of high-dimensional manifolds, as h-cobordant manifolds are often diffeomorphic.
With this in mind, given two h-cobordant manifolds M and M', how different can their diffeomorphism groups Diff(M) and Diff(M') be? The homotopy groups of these two spaces are the same “up to extensions” in a range of strictly positive degrees. Contrasting this fact, I will present examples of h-cobordant manifolds with different mapping class groups. In doing so, I will review the classical theory of h-cobordisms and introduce several moduli spaces of manifolds that help in answering this question
The Singer conjecture predicts concentration of L^2-Betti numbers for closed aspherical manifolds. I'll discuss recent progress on this conjecture for Tomei manifolds and some Gromov-Thurston branched covers, exploiting the fact that these manifolds have virtually special fundamental groups. One consequence is a counterexample to Lueck's homology torsion growth conjecture. Joint work with Boris Okun and Kevin Schreve.
The class of (finitely generated free)-by-cyclic groups has been extensively studied over the last few decades. The connections between the geometric and algebraic aspects of such groups and the dynamical properties of automorphism groups of finitely generated free groups make their study deep and interesting. The more general class of virtually free-by-cyclic groups, where here the free part is not necessarily finitely generated, is a less understood, but much more vast, class of groups. For example, it includes all surface groups, three-manifold groups of cohomological dimension two and one-relator groups with torsion. Allowing the free group to be infinitely generated presents extra challenges, but nevertheless leads to an interesting theory. In this talk I will discuss some new results on the geometry of free-by-cyclic groups. I will mostly focus on characterising when a free-by-cyclic group is locally quasi-convex and show how such a characterisation also leads to a characterisation of locally quasi-convex one-relator groups.
Coarse characterisations of planarity in Cayley graphs
Recall a graph is said to be planar if it can be drawn in the plane without edges crossing. Let's call a finitely generated group G (virtually) planar if (some finite-index subgroup of) G admits a planar Cayley graph.
In this talk I will present several results characterising virtually planar groups in terms of their coarse geometry, illustrating the philosophy that this class of groups is geometrically “very rigid”. For example, we will see how any finitely generated group which is QI to some planar graph must itself be virtually planar. Time permitting, I will also advertise some open problems in this direction.
The non-positive immersion property for generalized Wirtinger presentations
In the last decades, locally indicable groups have emerged in many works related to known problems in topology, algebra, and geometry. These groups are characterized by the property that every non-trivial finitely generated subgroup admits an epimorphism onto the integers. In particular, they appear in problems concerning asphericity, orderability, and equations over groups.
In the early 2000s, Wise introduced a topological variant of non-positive curvature for 2-complexes: the non-positive immersion property. This notion is closely related to local indicability and is also used to study the coherence and cohomological properties of the fundamental groups of 2-complexes. Recently, this concept has regained relevance due to its deep connection with Bausmlag's conjecture, which was recently proven by A. Jaikin-Zapirain and M. Linton.
In this talk, I will present new methods, developed in joint work with Gabriel Minian, to study the local indicability of groups that admit generalized Wirtinger presentations. We then apply our methods to study the non-positive immersion property for the 2-complexes associated to these presentations. This provides a large family of concrete examples of 2-complexes (which can be described algorithmically) with the non-positive immersion property.
The Dehn functions of subgroups in a direct product of free groups
Subgroups in a direct product of free groups form a widely-studied family of groups, whose algebraic structure is strongly related to the finiteness properties that they satisfy. We investigate the Dehn function of such groups; this is an algebraic invariant which represents the complexity of solving the word problem. We show that, for subgroups of type $F_{n-1}$ inside a product of $n$ free groups, the Dehn functions have a uniform upper bound of $N^9$. We also prove that the Bridson-Dison group has Dehn function exactly $N^4$; the lower bound is proved through a new homotopical invariant encoded in braid groups.
Reducing rooted graphs while preserving their Higman-Thompson groups
The Higman-Thompson groups are dense subgroups of the almost automorphism groups of quasi-regular trees that are finitely presented and (virtually) simple. Their isomorphism problem was solved by Pardo using the Leavitt algebra. In this talk, I will focus on a generalization of Higman-Thompson groups onto unfolding trees of rooted directed graphs. I will showcase certain reductions of graphs that preserve the Higman-Thompson group of its unfolding tree. The proofs that these reductions preserve the Higman-Thompson groups use the graph monoid and the Leavitt path algebra of a graph. I will also demonstrate when such unfolding trees have finite automorphism orbits(i.e. are cocompact).
Sierpinski Carpets and relatively hyperbolic groups
The topology of a Bowditch boundary of a relatively hyperbolic group (with respect to a set of parabolic subgroups) has its importance, not only as an invariant for the group, but as a way to encode algebraic information about it, such as decompositions of the group as graphs of groups or if the group is a group of isometries of a manifold, for example. We present some examples of topological spaces that can appear as boundaries of relatively hyperbolic groups.
Whyburn showed that if we take a 2-sphere and remove an infinite collection of open disks satisfying some properties, then we get the 1-dimensional Sierpiński Carpet. After that, Cannon generalized it for n-dimensional Sierpiński Carpets. Recently, Tshishiku and Walsh gave another characterization of the Sierpiński Carpet: if we take a 2-sphere, remove a countable dense set and replace each point by a circle, then we get a 1-dimensional Sierpiński Carpet. We generalized their result for a n-dimensional Sierpiński Carpet. Then, we are able to show that some groups have a Bowditch boundary homeomorphic to the n-dimensional Sierpiński Carpet.
Galois cohomology of reductive groups over global fields
Let F be a number field (say, the field of rational numbers Q) or a p-adic field (say, the field of p-adic numbers Q_p), or a global function field (say, the field of rational functions of one variable over a finite field F_q). Let G be a connected reductive group over F (say, SO(n) ). One needs the first Galois cohomology set H^1(F,G) for classification problems in algebraic geometry and linear algebra over F. In the talk, I will give closed formulas for H^1(F,G) when F is as above, in terms of the algebraic fundamental group \pi_1(G) introduced by the speaker in 1998. All terms will be defined and examples will be given.
On groups with quadratic rational character values
A finite group is said to be rational or rationally valued if its character table is formed by rational numbers. Although rationality is a strong condition, their study is a classical topic in character theory because some relevant groups, as for example the symmetric groups, are rational.
During the last years several authors have considered groups for which each character value belongs to a quadratic extension of the rationals.
These have led to several classes of groups depending on whether it is required that the full character table is contained in a fixed quadratic extension of the rational, or this condition is required per rows or columns of the character table. The groups satisfying the condition per rows are known as quadratic rational groups and those satisfying the conditions per columns are called semi-rational.
In this talk we discuss relations between some of these conditions with special emphasis on uniformly semi-rational groups which are groups G satisfying the following conditions for some integer r relatively prime with the exponent of G: For every g in G, every generator of the cyclic group generated by g is conjugate in G to g or g^r.
Joint work with Marco Vergani.
Fundamental groups of plane curve complements
The fundamental group of the complement of a plane curve C in the complex projective plane is a powerful invariant which is sensitive to the type and position of singularities of C. In this talk, we'll give an overview of fundamental groups of plane curve complements, including the Zariski-van Kampen method for finding presentations of these groups. We'll also discuss recent progress on the topic (joint with José Ignacio Cogolludo Agustín) regarding curve complements whose fundamental group is a free product of cyclic groups.
Commensurators of free groups and free pro-p groups
The commensurator Comm(Gamma) of a group Gamma is the group of isomorphisms between finite index subgroups of Gamma modulo coincidence on a finite index subgroup. The commensurator Comm(G) of a profinite group G is defined similarly, except that we consider finite index open subgroups. The group Comm(G) plays a central role in the study of locally compact groups that are locally isomorphic to G. I will survey recent results notably regarding simplicity of some natural open subgroups of Comm(G) for G a finitely generated free pro-p group, and
also of some subgroups of Comm(F) for F a finitely generated free (abstract) group. Joint with Yiftach Barnea, Mikhail Ershov, Colin Reid, Matteo Vannacci, Thomas Weigel.
Beyond Brauer's Height Zero Conjecture
Brauer's height zero conjecture asserts that all the irreducible characters in a Brauer p-block have height zero if and only if the defect group of the block is abelian. The proof of the famous height zero conjecture of Brauer was finally completed in 2022. I will introduce the basic concepts of block theory, discuss several extensions of this conjecture and recent progress on them. This talk is based on work of Gunter Malle, Noelia Rizo, Mandi Schaeffer Fry and the speaker, along with various combinations of the four of us.
Self-similar finite p-groups
It has been customary to define self-similarity for a group G as the existence of both a subgroup H of G having finite index m in G, and of a homomorphism $f: H \to G$. Then the group G is self-similar provided the only normal subgroup K of G which is f-invariant subgroup (that is, $K^f \leq K$) is trivial. In this case, the triple (G,H,f) is called simple. Such a simple triple leads to a faithful representation of G as a state-closed group of automorphisms of a regular one-rooted tree of degree m. In this talk, we consider a prime p and a finite state-closed p-group G of automorphisms of a regular one-rooted tree of degree p. We present recent results on the structure of G and, in particular, when G admits an injective virtual endomorphism.
The sigma 1 and 2 invariants for Artin groups
The Sigma-invariants are geometric invariants that encode the homological and homotopical finiteness properties of all co-abelian subgroups, i.e. subgroups of the form G’<N<G. On the other hand, the family of Artin groups is an important family of groups that generalizes the families of free groups, free abelian groups, right angled-Artin groups and braid groups among others.
The aim of this talk is to present two conjectures on the structure of the sigma 1 and sigma 2 invariants for Artin groups and to prove that they are true for some particular subfamilies of those groups.
Random surfaces and systoles
The systole of a hyperbolic (or more generally Riemannian) surface is the length of its shortest closed geodesic. This invariant plays a role in many places in hyperbolic geometry. How large the systole can be as a function of the topology of the surface - a hyperbolic version of the Euclidean lattice packing density problem - is a notoriously difficult problem. I will speak about joint work with Mingkun Liu on random constructions of hyperbolic surfaces with large systoles.
Filtración de Johnson e invariantes de tipo finito
En esta charla hablaré de la relación entre la filtración de Johnson del grupo de Torelli de una superficie orientable y la teoría de invariantes de tipo finito de esferas de homología de dimensión 3. La relación entre estos dos objetos se hace mediante la teoría de esciciones de Heegaard de las variedades, que permite entre otras cosas ver un invariante por ejemplo de esferas de homología, como una función sobre el grupo de Torelli que satisface ciertas propiedades combinatorias naturales. Es un hecho conocido que cualquier esfera de homología de dimensión 3 puede ser construida utilizando mediante un elemento del grupo de Torelli o mediante un elemento de cualquiera de los 4 primeros pasos de la filtración. En la segunda parte de la charla explicaré como el estudio de los $2$-cociclos triviales sobre un subgrupo del grupo de Torelli permite demostrar que este fenómeno no es cierto en el quinto paso, y que este escalón viene detectado por un invariante explícito. Los resultados presentados son fruto un trabajo con R. Riba.
Representations of the symmetric group and regularisations
I will recall some results on (complex) representation theory of finite groups, with a focus on symmetric groups. Namely, there is a natural probability measure defined on the set of irreducible representations. We will then study the p-regularisation map on partitions, which was introduced by James in order to study the representations of the symmetric group when the ground field has characteristic p. Based on a limit shape result of Kerov–Vershik and Logan-Shepp, we will study the shape of the p-regularisation of a large random partition. Finally, we will mention a generalisation of the p-regularisation map, defined by Diego Millan Berdasco.
Coarse groups and related questions
The underlying theme of this talk is the following: geometric group theory got us used to only consider metric spaces up to quasi-isometry. What happens if we apply the same paradigm to the group axioms as well? Namely, define coarse groups as metric spaces equipped with operations that satisfy the group axioms up to bounded error (formally, these are the group objects in the category of coarse spaces). In this talk I will give an introduction to these constructs, illustrating aspects we know, and pointing at some of the many things we still do not know about them.
The cancellation property for projective modules over integral group rings
Let G be a finite group and let Z[G] denote the integral group ring. If two finitely generated projective Z[G]-modules P and Q are isomorphic after taking a direct sum with the free module Z[G], are they necessarily isomorphic? If so, we say that Z[G] has the cancellation property. This was studied extensively in the 1960s-80s by H. Jacobinski, A. Fröhlich and R. G. Swan, and has applications both in number theory and algebraic topology. However, a complete classification of the finite groups which have the cancellation property had remained out of reach.
In this talk, I will present a new cancellation theorem for projective Z[G]-modules and explain how this leads to an approach to complete the classification using only finite computation. By utilising recent computer calculations of W. Bley, T. Hofmann and H. Johnston, this leads a classification of when cancellation occurs among all finite groups G which have no quotient which is a binary tetrahedral group, binary octahedral group, or a binary icosahedral group.
Cubically presented groups, asphericity, and the Cohen-Lyndon property
We’ll explore the problem of finding effective models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes, we’ll discuss the framework -- cubical small-cancellation theory -- that provides the necessary tools to do so, and we’ll explain how this viewpoint allows us to compute the homology and cohomology of various examples.
Division and Localization on Groupoid Graded Rings
A groupoid is a small category in which every morphism is invertible, and as such, generalizes the concept of groups. The concept of a groupoid graded ring is similar to that of group graded ring. That is, there exists additive subgroups for each arrow in the groupoid, whose multiplication makes sense with the composition law of the arrows if they can be composed, or is equal to 0 otherwise. Different from group graded rings, groupoid graded rings do not need to be unital. We suppose that our graded rings are object unital, that is, for every object in the groupoid, there exists an idempotent in the ring which acts as unity for products with homogeneous elements of compatible degrees.
Division is studied with respect to the aforementioned idempotents. In this talk, we'll discuss recent progress regarding division and localization on object unital groupoid graded rings. In particular, we'll discuss a generalization of P.M. Cohn's results which characterize homomorphisms from a ring to a division ring, previously generalized by D. E. N. Kawai and J. Sanchez to the context of group graded rings.
Totally disconnected locally compact groups, accessibility and Euler-Poincaré characteristic
In the first part of the talk I will illustrate how the classical notion of accessibility for finitely generated groups carries over to the realm of compactly generated totally disconnected locally compact (= t.d.l.c.) groups. Then, by means of a new notion of Euler-Poincaré characteristic, I will discuss an accessibility result in the t.d.l.c. framework, under the assumption of rational discrete cohomological dimension = 1.
L^2 homology and fixed points of automorphisms
Many useful techniques in group theory have been originally developed for the purpose of studying subgroups of free groups and their ranks. A classical example of this type of questions is the following conjecture of Scott. Let F be the free group of finite rank n and let phi be an automorphism of F, then the subgroup of elements fixed by phi has rank at most n. This was settled by Bestvina and Handel by developing the analogue theory of train-track maps for free groups. We will discuss a new approach to this problem based on L^2-homology.
Generalized torsion elements in groups
In this talk we present some properties of generalized torsion elements in groups. Moreover, we try connect this ``new'' concept with the usual concept of torsion in some standard class of groups (e.g., nilpotent, FC-groups, abelian-by-finite). This presentation is mainly based in the following papers [1,2,3].
[1] R. Bastos and L. Mendonca. Generalized torsion elements in finitely presented groups, in preparation.
[2] R. Bastos, C. Schneider and D. Silveira. Generalized torsion elements in groups, Arch. Math. (Basel), 122, 121--131, 2024.
[3] G. Naylor and D. Rolfsen. Generalized torsion in knot groups. Canad. Math. Bull., 59(1):182--189, 2016.
Talks 2023-24
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.
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.
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.
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.
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.
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.
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.
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?
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).
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.
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.
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.
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).
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.
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)
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.
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.
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.
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).
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.
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.
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).
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.
Taro Sakurai (Chiba University)
Non-isomorphic 2-groups with isomorphic modular group algebras refined
The first counterexamples to the modular isomorphism problem were found by García-Lucas, Margolis and del Río in 2021. This talk presents new counterexamples with isomorphisms and explains a new classification of certain 2-groups which leads to this discovery. This is joint work in progress with Leo Margolis.
Taro Sakurai (Chiba University)
On the local-global correspondence of conjugacy classes
Let p be a prime and let G be a finite group with a Sylow p-subgroup P. It is known for decades that there is a bijection from the set of conjugacy classes of G whose size is not divisible by p to that of N_G(P). This talk presents a refinement of this local-global correspondence by taking divisibility of sizes into account. We also propose a local-global conjectures on conjugacy classes of finite p-solvable groups whose size is not divisible by a prime p.
Marta Lesniak (University of Gdansk)
Torsion normal generators of the mapping class group of a nonorientable surface
An element of a mapping class group of a nonorientable surface can be a normal generator of the group if we can find a specific pair of curves on the surface. To understand how a torsion element acts on curves, we use the theory of Macbeath about actions of groups on surfaces to construct fundamental domains of both the orbifold surface and the original surface. Then we use this understanding to find pairs of curves satisfying the hypothesis for torsion elements of varying forms.