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Enric Ventura

Publications and source records attributed to Enric Ventura.

At least 19 recordsLinked to original sources
arXiv2026

Additive subgroups of Q: logarithmic description and intersection configurations

We develop a logarithmic framework to study the lattice of subgroups of the additive group of rational numbers. By encoding positive rationals via their prime exponent sequences, one obtains a bijection between $\mathbb{Q}^+$ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of $(\mathbb{Q},+)$ through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in $\mathbb{Z}\cup \{- \infty\}$. Within this framework, subgroup inclusion, sum, product, and intersection admit simple coordinatewise descriptions, providing a transparent interpretation of the subgroup lattice and recovering several classical results such as the subgroup classification up to isomorphism. Exploiting this perspective, we obtain a complete characterization of the intersection configurations realizable in subgroups of $(\mathbb{Q},+)$. For rank configurations, realizability is characterized by the decreasing condition together with restrictions on the possible ranks and on the minimal zero sets, and, in the finite-support case, by an additional cardinality condition. For binary configurations, this reduces to saying that every decreasing configuration is realizable in the infinite-support case, whereas in the finite-support case the mentioned cardinality condition is required.

math.GR↗
arXiv2025

Twisted conjugacy in $BS(n, 1)$

In this article, we solve the twisted conjugacy problem for solvable Baumslag--Solitar groups $BS(n,1)$, i.e., we propose an algorithm which, given two elements $u,v \in BS(n,1)$ and an automorphism $φ\in \Aut(BS(n,1))$, decides whether $v=(wφ)^{-1} u w$ for some $w\in BS(n,1)$. Also we prove that the automorphism group $\Aut(BS(n,1))$ is orbit decidable -- given two words on the generators $u,v\in F(X)$, decide whether the corresponding elements $u,v\in G$ can be mapped to each other by some automorphism in $\Aut(BS(n,1))$.

math.GR↗
arXiv2025

Computing $H$-equations with 2-by-2 integral matrices

We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral invertible matrices: ''given $h_1,\ldots, h_r; g\in \operatorname{PSL}_2(\mathbb{Z})$, decide whether $g$ is algebraic over the subgroup $H=\langle h_1,\ldots ,h_r\rangle \leqslant \operatorname{PSL}_2(\mathbb{Z})$ (i.e., whether there exist a non-trivial $H$-equation $w(x)\in H*\langle x\rangle$ such that $w(g)=1$) and, in the affirmative case, compute finitely many such $H$-equations $w_1(x),\ldots ,w_s(x)\in H*\langle x\rangle$ further satisfying that any $w(x)\in H*\langle x\rangle$ with $w(g)=1$ is a product of conjugates of $w_1(x),\ldots ,w_s(x)$''. The same problem for square matrices of size 4 and bigger is unsolvable.

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↗
arXiv2023

The central tree property and algorithmic problems on subgroups of free groups

We study the average case complexity of the uniform membership problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.

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

Computation of endo-fixed closures in free-abelian times free groups

In this paper, we explore the behaviour of the fixed subgroups of endomorphisms of free-abelian times free (FATF) groups. We exhibit an algorithm which, given a finitely generated subgroup $\mathcal{H}$ of a FATF group $\mathcal{G}$, decides whether $\mathcal{H}$ is the fixed subgroup of some (finite) family of endomorphisms of $\mathcal{G}$ and, in the affirmative case, it finds such a family. The algorithm combines both combinatorial and algebraic methods.

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↗
arXiv2022

Probabilistic nilpotence in infinite groups

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+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 Folner 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. 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. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. 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, 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, generalising a result of Gallagher.

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↗
arXiv2021

Onto extensions of free groups

An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $Γ_{A'}(K)$ is a quotient of $Γ_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.

math.GR↗
arXiv2020

Degrees of compression and inertia for free-abelian times free groups

We introduce the concepts of degree of inertia, $\text{di}_G(H)$, and degree of compression, $\text{dc}_G(H)$, of a finitely generated subgroup $H$ of a given group $G$. For the case of direct products of free-abelian and free groups, we compute the degree of compression and give an upper bound for the degree of inertia. Imposing some technical assumptions to the supremum involved in the definition of degree of inertia, we introduce the notion called restricted degree of inertia, $\text{di}'_G(H)$, and, again for the case $\mathbb{Z}^m \times F_n$, we provide an explicit formula relating it to the restricted degree of inertia of its projection to the free part, $\text{di}'_{F_n}(Hπ)$.

math.GR↗
arXiv2019

Fixed subgroups and computation of auto-fixed closures in free-abelian times free groups

The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of $m,n$. We also study periodic points of endomorphisms of $Z^m\times F_n$, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of $Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real elements in $Z^m\times F_n$, contributing a modest step into the project of doing so for any right angled Artin group (as McCool did with respect to Whitehead's Theorem in the free context).

math.GR↗
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.

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