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Jordi Delgado

Publications and source records attributed to Jordi Delgado.

At least 19 recordsLinked to original sources
arXiv2026

Pullbacks and intersections in categories of graphs of groups

We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called $\mathbb{A}$-product of two morphisms into a graph of groups $\mathbb{A}$ -- a graph of groups which, within the appropriate categorical setting, captures the intersection of subgroups of the fundamental group of $\mathbb{A}$. We show that, in the category of pointed graphs of groups, pullbacks always exist and correspond precisely to pointed $\mathbb{A}$-products. In contrast, pullbacks do not always exist in the category of unpointed graphs of groups. However, when they do exist, and we show that it is the case, in particular, under certain acylindricity conditions, they are again closely related to $\mathbb{A}$-products. We trace, all along, the parallels with Stallings' classical theory of graph immersions and coverings, in relation to the study of the subgroups of free groups. Our results are useful for studying intersections of subgroups of groups that arise as fundamental groups of graphs of groups. As an example, we carry out an explicit computation of a pullback which results in a classification of the Baumslag--Solitar groups with the finitely generated intersection property.

math.GR↗
arXiv2026

The finitely generated intersection property in fundamental groups of graphs of groups

A group $G$ is said to satisfy the finitely generated intersection property (f.g.i.p.) if the intersection of any two finitely generated subgroups of $G$ is again finitely generated. The aim of this article is to understand when the fundamental group of a graph of groups has the f.g.i.p. Our main results are general criteria for the f.g.i.p. in graphs of groups which depend on properties of the vertex groups, properties of certain double cosets of the edge groups and the structure of the underlying graph. For acylindrical graphs of groups, we also obtain criteria for the strong f.g.i.p. (s.f.g.i.p.). Our results generalise classical results due to Burns and Cohen on the f.g.i.p. for amalgamated free products and HNN extensions. As a concrete application, we show that a graph of locally quasi-convex hyperbolic groups with virtually $\mathbb{Z}$ edge groups (for instance, a generalised Baumslag--Solitar group) has the f.g.i.p. if and only if it does not contain $F_2\times\mathbb{Z}$ as a subgroup. In addition, we show that this condition is decidable. The main tools we use are the explicit constructions of pullbacks of immersions into a graph of group, obtained by the authors in a previous paper, and a technical condition on coset interactions, introduced in this paper.

math.GR↗
arXiv2025

Orbit problems for free-abelian times free groups and related families

We prove that the Brinkmann Problems (BrP & BrCP) and the twisted-conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group Fn x Z^m. Furthermore, we prove the decidability of the two-sided Brinkmann conjugacy problem (2BrCP) for monomorphisms of FATF groups (and combine it with TCP) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

math.GR↗
arXiv2024

Stallings automata

This text aims to provide a self-contained, comprehensive, and reasonably detailed presentation of the theory of Stallings automata and some of its main applications.

math.GR↗
arXiv2024

Quotient-saturated groups

We introduce the new notion of quotient-saturation as a measure of the immensity of the quotient structure of a group. We present a sufficient condition for a finitely presented group to be quotient-saturated, and use it to deduce that non-elementary finitely presented subgroups of a hyperbolic group (in particular, non-elementary hyperbolic groups themselves) are quotient-saturated. Finally, we elaborate on the previous results to extend the scope of this property to finitely presented acylindrically hyperbolic groups.

math.GR↗
arXiv2024

Free-abelian by free groups: homomorphisms and algorithmic explorations

We obtain an explicit description of the endomorphisms of free-abelian by free groups together with a characterization of when they are injective and surjective. As a consequence we see that free-abelian by free groups are Hopfian and not coHopfian, and we investigate the isomorphism problem and the Brinkmann Problem for this family of groups. In particular, we prove that the isomorphism problem (undecidable in general) is decidable when restricted to finite actions, and that the Brinkmann Problem is decidable both for monomorphisms and automorphisms.

math.GR↗
arXiv2023

Intersection configurations in free and free times free-abelian groups

In this paper we study intersection configurations -- which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability -- in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots , H_k \leqslant G$ realizing it. It is well known that free groups $\mathbb{F}_n$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups $\mathbb{F}_n \times \mathbb{Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a `basis' for it. We finally prove that any intersection configuration is realizable in a FTFA group $\mathbb{F}_n \times \mathbb{Z}^m$, for $n\geq 2$ and big enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

math.GR↗
arXiv2023

Autòmats de Stallings, un camí d'anada i tornada

In this paper we review some of the fundamental properties of the free group and give a detailed account of Stallings's theory of automata, a geometric interpretation of its subgroups that has been (and still is) immensely fruitful, both as a means of understanding classical results, as as a source of new results. We review some of the most important ones. -- - En aquest article revisem algunes de les propietats fonamentals del group 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 entendre resultats clàssics, com com a font de nous resultats. N'expliquem alguns dels més rellevants.

math.GR↗
arXiv2022

A list of applications of Stallings automata

This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case.

math.GR↗
arXiv2022

Stallings automata for free-times-abelian groups: intersections and index

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively.

math.GR↗
arXiv2021

Relative order and spectrum in free and related groups

In this paper, we consider a natural generalization of the concept of order of an element in a group: an element $g \in G$ is said to have order $k$ in a subgroup $H$ of $G$ (\resp \wrt a coset $Hu$) if $k$ is the first strictly positive integer such that $g^k \in H$ (\resp $g^k \in Hu$). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (\ie the set of elements of a given order) \wrt finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even non-recursively enumerable sets of natural numbers. Also, we take advantage of Mikhailova's construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.

math.GR↗
arXiv2020

On the lattice of subgroups of a free group: complements and rank

A $\vee$-complement of a subgroup $H \leqslant \mathbb{F}_n$ is a subgroup $K \leqslant \mathbb{F}_n$ such that $H \vee K = \mathbb{F}_n$. If we also ask $K$ to have trivial intersection with $H$, then we say that $K$ is a $\oplus$-complement of $H$. The minimum possible rank of a $\vee$-complement (resp. $\oplus$-complement) of $H$ is called the $\vee$-corank (resp. $\oplus$-corank) of $H$. We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the $\vee$-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds.

math.GR↗
arXiv2019

Toward a Public MAGIC Gamma-Ray Telescope Legacy Data Portal

The MAGIC telescopes are one of the three major IACTs (Imaging Atmospheric Cherenkov Telescopes) for observation of gamma rays in the TeV regime currently operative. MAGIC functions since 2003, and has published data from more than 60 sources, mostly blazars. MAGIC already provides astronomical \texttt{.fits} files with basic final scientific products such as spectral energy distributions, light curves and skymaps from published results. In future, the format of the files can be complemented with further relevant information to the community: a) by including the full multi-wavelength dataset enclosed in a publication, b) providing data in alternative easy-to-use formats such as ASCII or ECSV, which are accessible with other commonly used packages such as \texttt{astropy} or \texttt{gammapy}. Finally, besides high level products, activities have started to provide photon event lists and instrument response functions in a format such that scientists within and outside the community are allowed to perform higher level analysis. A second aim is to provide a full legacy of MAGIC data. This contribution will illustrate the achievements and plans of this activity.

astro-ph.IM↗
arXiv2018

Intersection problem for Droms RAAGs

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 $H \cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H \cap K$. 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.

math.GR↗
arXiv2016

Algorithmic recognition of infinite cyclic extensions

We prove that one cannot algorithmically decide whether a finitely presented $\mathbb{Z}$-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique $\mathbb{Z}$-extensions, and the semi-conjugacy problem for deranged outer automorphisms.

math.GR↗
arXiv2014

Some characterizations of Howson PC-groups

We show that in the class of partially commutative groups, the conditions of being Howson, being fully residually free, and being free product of free-abelian groups, are equivalent.

math.GR↗
arXiv2014

Stopping Criteria in Contrastive Divergence: Alternatives to the Reconstruction Error

Restricted Boltzmann Machines (RBMs) are general unsupervised learning devices to ascertain generative models of data distributions. RBMs are often trained using the Contrastive Divergence learning algorithm (CD), an approximation to the gradient of the data log-likelihood. A simple reconstruction error is often used to decide whether the approximation provided by the CD algorithm is good enough, though several authors (Schulz et al., 2010; Fischer & Igel, 2010) have raised doubts concerning the feasibility of this procedure. However, not many alternatives to the reconstruction error have been used in the literature. In this manuscript we investigate simple alternatives to the reconstruction error in order to detect as soon as possible the decrease in the log-likelihood during learning.

cs.LG↗