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El tema central d'aquest treball és l'invariant Bieri-Neuman-Streble (BNS) per grups finitament generats. Es tracta d'un concepte que relaciona l'àlgebra abstracta amb la geometria, ja que és un subconjunt obert de l'esfera unitària de $\mathbb{R}^n$, per una certa $n$ relacionada amb el grup. El definirem en termes de la connectivitat de certs subgrafs de Cayley i estudiarem algunes de les seves propietats algebraiques. També el relacionarem amb conceptes i resultats moderns d'indecidibilitat per a grups, concloent que no és possible decidir la pertinença o no d'un punt donat a l'invariant.

Aquest treball tracta del grau de commutativitat de grups no abelians finitament generats. Provarem al primer capítol un resultat clàssic de Gustafson per al grau de commutativitat de grups finits, en concret veurem que el grau de commutativitat està fitat per 5/8 per a grups finits no abelians. Al segon capítol farem la construcció del grup lliure de cardinal arbitrari i introduirem el concepte de presentació d'un grup, per tal de poder, al tercer capítol, traslladar la noció de grau de commutativitat a grups finitament generats. Finalment, també veurem que per a grups finitament generats de creixement subexponencial i residualment finits, si el grup és no abelià aleshores el seu grau de commutativitat està fitat també per 5/8, i que si el seu grau de commutativitat és positiu, aleshores és un grup virtualment abelià.

L'objectiu d'aquesta tesi és estudiar les representacions de GL2(Fp). Subdividim la nostra presentació en tres capítols. El primer dóna una visió general bàsica de la teoria de representació i les eines necessàries per construir els tipus de representacions més senzills. Les més difícils les creem a partir del que es coneix com la representació de Weil. Això implica alguns conceptes de cohomologia de grups, que s'expliquen al Capítol 2 i ocupen la major part de l'exposició. Finalment, en l'últim capítol, acabem la construcció de les representacions., El objetivo de esta tesis es estudiar las representaciones de GL2(Fp). Dividimos nuestra presentación en tres capítulos. El primero proporciona una visión general básica de la teoría de representación y las herramientas necesarias para construir los tipos más simples de representaciones. Las más difíciles las creamos a partir de lo que se conoce como la representación de Weil. Esto implica algunos conceptos de cohomología de grupos, que se explican en el Capítulo 2 y ocupan la mayor parte de la exposición. Finalmente, en el último capítulo completamos la construcción de las representaciones., The aim of this thesis is to study the representations of GL2(Fp). We subdivide our presentation into three chapters. The first gives a basic overview of representation theory and the necessary tools to construct the easier types of representations. We make the harder ones out of what is known as the Weil representation. This involves some group cohomology concepts, which are explained in Chapter 2 and take the largest portion of the exposition. Finally, in the last chapter we complete the construction of the representations., Outgoing

A Moore (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in- and outdegree z, diameter k, and number of vertices (or order) attaining the corresponding Moore bound M(r, z, k) for mixed graphs. When the order of G is close to M(r, z, k) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z, 3)-mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph., This research has been partially supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RBI00., Peer Reviewed, Postprint (published version)

L'estudi dels subgrups del grup lliure es pot dur a terme utilitzant la teoria desenvolupada per Stallings, que dóna una representació gràfica que codifica les propietats algebraiques d'aquests subgrups en forma d'autòmats de Stallings. Aquests autòmats es poden utilitzar per resoldre molts problemes algorítmics per al grup lliure, com el problema de la pertinença, el problema de la intersecció o el problema de l'índex finit. En aquest treball, oferim una visió general de la teoria de Stallings i l'apliquem per demostrar un resultat de Hall que dóna el nombre de subgrups d'índex finit al grup lliure. A més, presentem una generalització de la teoria de Stallings que dóna una representació geomètrica dels subgrups dels grups lliures per lliure-abelians mitjançant autòmats de Stallings enriquits amb cert etiquetatge abelià per considerar la part lliure-abeliana d'aquests grups. Combinant aquesta teoria amb el resultat de Hall i un resultat anàleg per a grups lliure-abelians, obtenim una fórmula recursiva que dóna el nombre de subgrups d'índex finit en un grup lliure per lliure abelià., El estudio de los subgrupos del grupo libre puede llevarse a cabo utilizando la teoría desarrollada por Stallings, en en la que se da una representación gráfica que codifica las propiedades algebraicas de estos subgrupos en forma de autómatas de Stallings. Estos autómatas pueden utilizarse para resolver muchos problemas algorítmicos para grupo libre, como el problema de la pertenencia, el problema de la intersección o el problema del índice finito. En este trabajo, ofrecemos una visión general de la teoría de Stallings y la aplicamos para demostrar un resultado de Hall que da el número de subgrupos de índice finito en el grupo libre. Además, presentamos una generalización de la teoría de Stallings que da una representación geométrica de los subgrupos de los grupos libres por libre-abelianos mediante autómatas de Stallings enriquecidos con cierto etiquetado abeliano para dar cuenta de la parte libre-abeliana de estos grupos. Combinando esta teoría con el resultado de Hall y un resultado análogo para grupos libre-abelianos, obtenemos una fórmula recursiva que da el número de subgrupos de índice finito en un grupo libre por libre-abeliano., The study of subgroups of the free group can be carried out using the theory developed by Stallings, in which a graphical representation encoding the algebraic properties of these subgroups is given in the form of Stallings automata. These automata can be used to solve many algorithmic problems for the free group, like the membership problem, the intersection problem or the finite index problem. In this work, we provide an overview of Stallings theory and apply it to prove a result by Hall that gives the number of finite index subgroups in the free group. Moreover, we present a generalisation of the theory of Stallings which gives a geometric representation of the subgroups of free times free-abelian groups using Stallings automata enriched with certain abelian labelling to account for the free-abelian part of these groups. Combining this theory with the result by Hall and an analogous result for free-abelian groups, we obtain a recursive formula giving the number of finite index subgroups in a free times free-abelian group.

The goal of this Advanced Graduate Course is to introduce the student into the world of free groups, to show the intrinsic complexity of this algebraic structure and of the natural problems emanating from it, and to introduce the modern Stallings techniques, able to solve most of the classical problems and many new ones in a quite comprehensible and graphical way. We’ll emphasize the computational point of view, not just giving formal proofs, but also providing algorithms able to do the tasks effectively. The course will cover several classical results, like Nielsen-Schreier Theorem, Schreier index formula, Marshall Hall Theorem, membership problem, residual finiteness, the Howson property, Hanna-Neumann inequality, etc. But we plan to also introduce more advanced material in direct connection with research done in the last years: techniques for giving asymptotic estimates of properties of subgroups of the free group, enriched Stallings graphs allowing to extend results to broader families., The theory of Stallings automata provides a neat geometric representation of subgroups of the free group, and constitutes the modern —and probably the most natural and fruitful — approach to their study. Moreover, if the involved subgroups are finitely generated, then this description is finitary and very well suited for algorithmic treatment. The original result (hinted by the work of Serre, and stated in a seminal paper bytallings in 1983) interprets the subgroups of a free group FA as covering spaces of the bouquet of n circles. This mainly topological original viewpoint, has been converted into a more combinatorial one during the last decades, with the use of automata; this stresses the fact that, when restricted to finitely generated subgroups, most of the interesting problems can be resolved in a computational way, with algorithms usually more efficient and easy to understand than classical ones (more algebraic oriented). The goal of this Advanced Graduate Course is to introduce the student into the world of free groups, to show him/her the intrinsic complexity of this algebraic structure and of the natural problems emanating from it, and to introduce him/her into the modern Stallings techniques, able to solve most of the classical problems and many new ones in a quite comprehensible and graphical way (and not using much technicalities). We’ll continuously emphasise the computational point of view, not just giving formal mathematical proofs, but also providing algorithms able to do the tasks effectively by using present time computers. On one hand, this is a research topic relatively easy to get in, since the background needed is just some basic knowledge of algebra and graph theory. On the other hand, it is a hot research topic with lots of papers published since Stallings seminal paper in 1983 using these techniques, and lots that continue appearing in our days. A significant part of the publications of the three proposed lecturers use Stallings graphical te, Postprint (published version)

L'enfocament principal d'aquest projecte se centra en el grup de Baumslag i Solitar (BS) i la relació amb la funció de Dehn. El nostre objectiu fonamental és investigar l'estructura i la presentació del grup BS, així com aprofundir en el concepte de la funció de Dehn. Posteriorment, demostrarem que el grup de Baumslag i Solitar exhibeix un creixement exponencial en la funció de Dehn. La primera secció proporcionarà els fonaments de teoria de grups rellevants pel projecte i presentarà el grup BS. A la segona secció, introduirem conceptes clau de teoria geomètrica de grups, com el graf de Cayley i els CW-complexos, que seran fonamentals per comprendre la funció de Dehn. A més a més, presentarem la definició d'aquesta funció en aquesta secció. Finalment, a la tercera secció, aprofundirem en la funció de Dehn, presentarem resultats addicionals i completarem la demostració del nostre objectiu principal., El enfoque principal de este proyecto se centra en el grupo de Baumslag y Solitar (BS) y su relación con la función de Dehn. Nuestro objetivo fundamental es investigar la estructura y presentación del grupo BS, así como profundizar en el concepto de la función de Dehn. Posteriormente, demostraremos que el grupo de Baumslag y Solitar exhibe un crecimiento exponencial en su función de Dehn. La primera sección proporcionará los fundamentos de teoría de grupos relevantes para el proyecto y presentará el grupo BS. En la segunda sección, introduciremos conceptos clave de teoría geométrica de grupos, como el grafo de Cayley y los CW-complejos, que serán fundamentales para comprender la función de Dehn. Además, presentaremos la definición de esta función en esta sección. Finalmente, en la tercera sección, profundizaremos en la función de Dehn, presentaremos resultados adicionales y completaremos la demostración de nuestro objetivo principal., The main focus of this project is on the Baumslag and Solitar (BS) group and its relationship with the Dehn function. Our fundamental objective is to investigate the structure and presentation of the BS group, as well as to delve into the concept of the Dehn function. Subsequently, we will show that the Baumslag and Solitar group exhibits exponential growth in its Dehn function. The first section will provide the group theory foundations relevant to the project and introduce the BS group. In the second section, we will introduce key concepts of geometric group theory, such as the Cayley graph and CW-complexes, which will be fundamental to understanding Dehn's function. Furthermore, we will present the definition of this function in this section. Finally, in the third section, we will delve deeper into the Dehn function, present additional results, and complete the proof of our main objective.

En aquest article revisem algunes de les propietats fonamentals del grup lliure i fem una exposició detallada de la teoria dels autòmats de Stallings, una interpretació geomètrica dels seus subgrups que ha estat (i segueix essent) immensament fructífera, tant com a mitjà per a entendre resultats clàssics com com a font de nous resultats. N’expliquem alguns dels més rellevants., Els autors agraïm el suport parcial rebut de l’Agencia Estatal de Investigación, a través del projecte de recerca MTM2017-82740-P (AEI/FEDER, UE). El primer autor va realitzar la primera part d’aquest treball a la Universitat del País Basc (UPV/EHU) amb suport parcial del MINECO a través del projecte PID2019- 107444GA-I00, i del Govern basc amb el projecte IT974-16., Postprint (published version)

In this paper, we consider the T - and V -versions, Tt and Vt , of the irrational slope Thompson group Ft considered in J. Burillo, B. Nucinkis and L. Reeves [An irrational-slope Thompson's group, Publ. Mat. 65 (2021), 809–839]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for T and V . We also show that Tt and Vt have index- 2 normal subgroups, unlike their original Thompson counterparts T and V . These index- 2 subgroups are shown to be simple., Peer Reviewed, Postprint (published version)

The version of record is available online at: http://dx.doi.org/10.1007/s11856-021-2168-3 The first and third authors gratefully acknowledge the grant SEV-20150554 which partially funded their stay at the ICMAT, where a part of the research for this paper was done. The second author was supported by grant FN 200021_163417/1 of the Swiss National Fund for scientific research., The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x1,…, xk+1 ¿ Gk+1 for which the simple commutator [x1, …, xk+1] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Følner sequence if G is amenable. In our first main result we show that if G is finitely generated, then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case k = 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ¿ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ¿ is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below, then G is virtually k-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher., Peer Reviewed, Postprint (author's final draft)

Es realitza un estudi de les extensions algebraiques de subgrups lliures, fent ús extensiu dels grafs de Stallings, en àmbits com la sobrejectivitat de morfismes entre grafs de Stallings, o el comportament d'aquestes com a reticle. Es revisa una conjectura de Miasnikov-Ventura-Weil, els contraexemples presentats per Parzanchevski-Puder i Kolodner, i es demostra una reformulació de la conjectura original. Continuant aquestes deliberacions, es proposen nous tipus d'extensions (injectives i sobrejectives) amb comportaments molt similars a les extensions algebraiques i factors lliures, conjecturant finalment si alguns resultats rellevants sobre les darreres es poden extentre a les primeres. Per a tal de plantejar aquestes qüestions, es fa una intruducció detallada sobre els grafs de Stallings i es presenten resultats clàssics sobre aquests i les seves aplicacions algebraiques de manera extensiva.

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0, In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that every partition of the number n gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs and to a recent known family of mixed graphs (having edges with and without direction). As a byproduct, the existence of perfect codes in allows us to give a lower bound for the multiplicity of its eigenvalue -1., Peer Reviewed, Postprint (author's final draft)

L'objectiu d'aquest treball és trobar explícitament un subgrup de les matrius 4x4 al que, 2 vectors u i v qualsevols, no es pot decidir l'existència o no d'una matriu A del subgrup tal que Au = v, un problema que podria semblar d'àlgebra lineal però que acaba sent molt més complicat. Aquest resultat es basa en un article de O. Bogopolski, A. Martino i E. Ventura on es va demostrar l'existència d'un subgrup amb les propietats que volem. A partir d'aquest resultat i d'un grup explícit donat per Collins, i resseguint els càlculs realitzats en els diferents articles citats a les referències, podem trobar i llistar les matrius desitjades.

We give an explicit characterization of which direct products G of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of G is compressed, and of which is it always inert., Peer Reviewed, Postprint (author's final draft)

OD matrices that describe mobility patterns provide major input to most transport analysis models. Since OD matrices are not yet directly observable, they are usually estimated indirectly. Data from new sources such as mobile phone records and GPS traces from mobile apps are emerging alternatives that allow for cheaper and timely estimates. However, they are still hindered by weaknesses that must be studied for use as input to transportation models. This paper presents a case study using mobility data from mobile phone records, with the goal of establishing a methodology for validating the obtained OD matrices and generating the appropriate input for traffic assignment models. Spatial and temporal consistency of OD data elaborated from mobile records has been proved to be useful for transportation modelling needs., Peer Reviewed, Postprint (published version)

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K of G, decides whether HnK is finitely generated or not, and, in the affirmative case, it computes a set of generators for HnK. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. F_2 x F_2) even have unsolvable SIP., Peer Reviewed, Preprint

This is a list of problems in combinatorial number theory gathered on the occasion of the meeting The Music of Numbers to honor the memory of the late Javier Cilleruelo (1961-2016)., Postprint (author's final draft)

For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of seed by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper is devoted to presenting the results which are related to this approach, leading to a new algorithm for computing and counting the semigroups of a given genus., Postprint (author's final draft)

For a curve of genus > 1 defined over a finite field, we present a sufficient criterion for the non-existence of automorphisms of order a power of a rational prime. We show how this criterion can be used to determine the automorphism group of some modular curves of high genus., Postprint (author's final draft)