arXiv subjects

Jone Lopez de Gamiz Zearra

Publications and source records attributed to Jone Lopez de Gamiz Zearra.

10 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.

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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.

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arXiv2025

Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case

We generalize to (certain) Artin groups some results previously known for right-angled Artin groups (RAAGs). First, we generalize a result by Droms, B. Servatius, and H. Servatius, and prove that the derived subgroup of an Artin group is free if and only if the group is coherent. Second, coherent Artin groups over non complete graphs split as free amalgamated products along free abelian subgroups, and we extend to arbitrary Artin groups admitting such a splitting a recent result by Casals-Ruiz and the first author on finitely generated normal subgroups of RAAGs. Finally, we use splittings of even Artin groups of FC-type to generalize results of Minasyan and Osin on acylindrical hyperbolicity of their subgroups.

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arXiv2025

Separability properties of higher-rank GBS groups

A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. In this paper we classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable and cyclic subgroup separable.

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arXiv2023

Finitely presented subgroups of direct products of graphs of groups with free abelian vertex groups

A result by Bridson, Howie, Miller, and Short states that if $S$ is a finitely presented subgroup of the direct product of free groups, then $S$ is virtually a nilpotent extension of a direct product of free groups. Moreover, if $S$ is a subgroup of type $FP_n$ of the direct product of $n$ free groups, then the nilpotent extension is finite, so $S$ is actually virtually the direct product of free groups. In this paper, these results are generalized to $2$-dimensional coherent right-angled Artin groups. More precisely, we show that a finitely presented subgroup of the direct product of $2$-dimensional coherent RAAGs is still virtually a nilpotent extension of a direct product of subgroups. If $S$ is moreover a type $FP_n$ subgroup of the direct product of $n$ $2$-dimensional coherent RAAGs, then $S$ is commensurable to a kernel of a character of a direct product of subgroups. Finally, we show that the multiple conjugacy problem and the membership problem are decidable for finitely presented subgroups of direct products of $2$-dimensional coherent RAAGs.

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arXiv2023

Subgroups of the direct product of graphs of groups with free abelian vertex groups

A result of Baumslag and Roseblade states that a finitely presented subgroup of the direct product of two free groups is virtually a direct product of free groups. In this paper we generalise this result to the class of cyclic subgroup separable graphs of groups with free abelian vertex groups and cyclic edge groups. More precisely, we show that a finitely presented subgroup of the direct product of two groups in this class is virtually $H$-by-(free abelian), where $H$ is the direct product of two groups in the class. In particular, our result applies to 2-dimensional coherent right-angled Artin groups and residually finite tubular groups. Furthermore, we show that the multiple conjugacy problem and the membership problem are decidable for finitely presented subgroups of the direct product of two $2$-dimensional coherent RAAGs.

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arXiv2023

On subdirect products of type $FP_n$ of limit groups over Droms RAAGs

We generalize some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if $S$ is a full subdirect product of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of $S$ to the direct product of any $s$ of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to $m$. In particular, this gives the values of the analytic $L^2$-Betti numbers of these groups in the respective dimensions.

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arXiv2022

On finitely generated normal subgroups of right-angled Artin groups and graph products of groups

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we deduce, among others, that finitely generated normal subgroups of RAAGs have decidable word, conjugacy and membership problems and that they are hereditarily conjugacy separable.

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arXiv2021

On the Bieri-Neumann-Strebel-Renz invariants and limit groups over Droms RAAGs

For a group $G$ that is a limit group over Droms RAAGs such that $G$ has trivial center, we show that $Σ^1(G) = \emptyset = Σ^1(G, \mathbb{Q})$. For a group $H$ that is a finitely presented residually Droms RAAG we calculate $Σ^1(H)$ and $Σ^2(H)_{dis}$. In addition, we obtain a necessary condition for $[χ]$ to belong to $Σ^n(H)$.

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arXiv2020

Subgroups of direct products of limit groups over Droms RAAGs

A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG and that their finitely presentable subgroups are separable.

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