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Montserrat Casals-Ruiz

Publications and source records attributed to Montserrat Casals-Ruiz.

At least 19 recordsLinked to original sources
arXiv2025

On the isomorphism problem for generalized Baumslag-Solitar groups: angles

We introduce a new isomorphism invariant for generalized Baumslag-Solitar (GBS) groups, which we call the limit angle. Unlike previously known invariants, which are primarily algebraic, the limit angle admits a dynamical interpretation, arising exclusively in the case of two interacting edges. This invariant captures subtle geometric behavior that does not manifest in configurations with more interacting edges. As an application, we use the limit angle to obtain a classification of GBS groups with one vertex and two edges.

math.GR↗
arXiv2025

On the isomorphism problem for cyclic JSJ decompositions: vertex elimination

We introduce two new moves on graphs of groups with cyclic edge groups that preserve the fundamental group. These moves allow us to address the isomorphism problem without the use of expansions, therefore keeping the number of vertices and edges constant along sequences of moves witnessing an isomorphism between two groups. We further show that the isomorphism problem for a large family of cyclic JSJ decompositions reduces to the case of generalized Baumslag-Solitar groups (GBS), and that among GBSs, it suffices to consider one-vertex graphs. As an application of the methods, we solve the isomorphism problem for a broad class of flexible GBSs. Finally, we discuss potential further applications of our techniques.

math.GR↗
arXiv2025

Automorphisms of graphs of groups with cyclic edge groups

We describe the outer automorphism group of a one-ended fundamental group of a graph of groups, when edge groups are cyclic, and vertex groups are torsion-free with cyclic centralizers. We show that in this case the outer automorphism group is virtually built from the outer automorphisms of the vertex groups (fixing some elements), the outer automorphisms of some associated generalized Baumslag-Solitar groups, and generalized twists - partial conjugations by elements in the centralizers of some elliptic elements in the group.

math.GR↗
arXiv2024

Presentation of kernels of rational characters of right-angled Artin groups

In this note, we characterise when the kernel of a rational character of a right-anlged Artin group, also known as generalised Bestiva-Brady group, is finitely generated and finitely presented. In these cases, we exhibit a finite generating set and a presentation. These results generalise Dicks and Leary's presentations of Bestina-Brady kernels and provide an algebraic proof for the results proven by Meier, Meinert, and VanWyk.

math.GR↗
arXiv2023

Pro-$\mathcal{C}$ RAAGs

Let $\mathcal{C}$ be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-$\mathcal{C}$ group $G_Γ$ (pro-$\mathcal{C}$ RAAG for short) is the pro-$\mathcal{C}$ completion of the right-angled Artin group $G(Γ)$ associated with the finite simplicial graph $Γ$. In the first part, we describe structural properties of pro-$\mathcal{C}$ RAAGs. Among others, we describe the centraliser of an element and show that pro-$\mathcal{C}$ RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-$p$ subgroups of pro-$\mathcal{C}$ RAAGs are either free pro-$p$ or free abelian pro-$p$. In the second part, we characterise splittings of pro-$\mathcal{C}$ RAAGs in terms of the defining graph. More precisely, we prove that a pro-$\mathcal{C}$ RAAG $G_Γ$ splits as a non-trivial direct product if and only if $Γ$ is a join and it splits over an abelian pro-$\mathcal{C}$ group if and only if a connected component of $Γ$ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-$\mathcal{C}$ RAAG, in the sense of Guirardel and Levitt.

math.GR↗
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.

math.GR↗
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.

math.GR↗
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.

math.GR↗
arXiv2022

Limit groups over coherent right-angled Artin groups

A new class of groups $\mathcal{C}$, containing all coherent RAAGs and all toral relatively hyperbolic groups, is defined. It is shown that, for a group $G$ in the class $\mathcal{C}$, the $\mathbb{Z}[t]$-exponential group $G^{\mathbb{Z}[t]}$ may be constructed as an iterated centraliser extension. Using this fact, it is proved that $G^{\mathbb{Z}[t]}$ is fully residually $G$ (i.e. it has the same universal theory as $G$) and so its finitely generated subgroups are limit groups over $G$. If $\mathbb{G}$ is a coherent RAAG, then the converse also holds - any limit group over $\mathbb{G}$ embeds into $\mathbb{G}^{\mathbb{Z}[t]}$. Moreover, it is proved that limit groups over $\mathbb{G}$ are finitely presented, coherent and CAT$(0)$, so in particular have solvable word and conjugacy problems.

math.GR↗
arXiv2021

On the elementary theory of graph products of groups

In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition which generalizes a range of natural ``non-freeness conditions" such as the satisfaction of a group law, having nontrivial center or being boundedly simple. As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups to describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results on the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent or classical linear groups.

math.GR↗
arXiv2020

Two Remarks on First-Order Theories of Baumslag-Solitar Groups

In this note we characterise all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS$(1,n)$. It turns out that a finitely generated group $G$ is elementarily equivalent to BS$(1,n)$ if and only if $G$ is isomorphic to BS$(1,n)$. Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.

math.GR↗
arXiv2019

On the positive theory of groups acting on trees

We study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, i.e. coincides with the positive theory of a non-abelian free group. Our results apply to a wide class of groups, including non-virtually solvable fundamental groups of $3$-manifold groups, generalised Baumslag-Solitar groups and almost all one-relator groups and graph products of groups. It follows that groups in the class satisfy a number of algebraic properties: for instance, their verbal subgroups have infinite width and, although some groups in the class are simple, they cannot be boundedly simple. In order to prove these results we describe a uniform way for constructing (weak) small cancellation tuples from (weakly) stable elements. This result of interest in its own is fundamental to obtain corollaries of general nature such as a quantifier reduction for positive sentences or the preservation of the non-trivial positive theory under extensions of groups.

math.GR↗
arXiv2019

Commensurability of Baumslag-Solitar groups

In this paper we classify Baumslag-Solitar groups up to commensurability. In order to prove our main result we give a solution to the isomorphism problem for a subclass of Generalised Baumslag-Solitar groups.

math.GR↗
arXiv2018

On commensurability of right-angled Artin groups I: RAAGs defined by trees of diameter 4

In this paper we study the classification of right-angled Artin groups up to commensurability. We characterise the commensurability classes of RAAGs defined by trees of diameter 4. In particular, we prove a conjecture of Behrstock and Neumann that there are infinitely many commensurability classes. Hence, we give first examples of RAAGs that are quasi-isometric but not commensurable.

math.GR↗
arXiv2018

On commensurability of some right-angled Artin groups II: RAAGs defined by paths

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length $n>4$ is commensurable to a RAAG defined by a tree of diameter 4 if and only if $n$ is 2 modulo 4. These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

math.GR↗