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 the membership problem is decidable for their finitely presented subgroups

Subgroups of direct products of limit groups over coherent right-angled Artin groups

xiii, 160 p.During the last 40 years, a large body of work has been directed to study the connection betweenfiniteness properties of groups and their algebraic and algorithmic properties. One of the early results isdue to Baumslag and Roseblade, who showed that while finitely generated subgroups of the directproduct of two free groups are wild and untractable, the finitely presented ones have nice algebraic andalgorithmic properties. This work was widely extended during the years to the class of finitely presentedresidually free groups viewed as subgroups of direct products of limit groups.In this thesis we continue this study and show that the good behaviour of finitely presented subgroupsextends to the class of finitely presented residually Droms RAAGs. More precisely, we give a completecharacterisation of finitely presented residually Droms RAAGs and we obtain a number of consequencesrelated to decision problems, growth of homology and the Bieri-Neumann-Strebel-Renz invariants. Wealso study the subgroup structure of direct products of limit groups over Droms RAAGs depending ontheir finiteness properties. Finally, we initiate the study of finitely presented subgroups of direct productsof $2$-dimensional coherent RAAGs

Finite graphs, free groups and Stallings’ foldings

[EN] The main goal of the work is to study some basic properties of free groups, by using group theory and topology. For this, graphs play a main role, since we will prove that the fundamental group of a connected graph is a free group. Therefore, basic notions and properties of free groups and graphs are explained firstly. Then, it is established a relation between operations in the category of graphs (the pullback and the pushout) and group theoretic operations between subgroups of free groups (the intersection and the join). After that, it is developed the theory of coverings of graphs. Finally, it is given an algorithm in order to decide whether a finitely generated subgroup of a free group is of finite index

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 FPn(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

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

Subgroups of direct products of limit groups over coherent right-angled Artin groups

During the last 40 years, a large body of work has been directed to study the connection between finiteness properties of groups and their algebraic and algorithmic properties. One of the early results is due to Baumslag and Roseblade, who showed that while finitely generated subgroups of the direct product of two free groups are wild and untractable, the finitely presented ones have nice algebraic and algorithmic properties. This work was widely extended during the years to the class of finitely presented residually free groups viewed as subgroups of direct products of limit groups. In this thesis we continue this study and show that the good behaviour of finitely presented subgroups extends to the class of finitely presented residually Droms RAAGs. More precisely, we give a complete characterisation of finitely presented residually Droms RAAGs and we obtain a number of consequences related to decision problems, growth of homology and the Bieri-Neumann-Strebel-Renz invariants. We also study the subgroup structure of direct products of limit groups over Droms RAAGs depending on their finiteness properties. Finally, we initiate the study of finitely presented subgroups of direct products of 2-dimensional coherent RAAGs

Finite graphs, free groups and Stallings’ foldings

[EN] The main goal of the work is to study some basic properties of free groups, by using group theory and topology. For this, graphs play a main role, since we will prove that the fundamental group of a connected graph is a free group. Therefore, basic notions and properties of free groups and graphs are explained firstly. Then, it is established a relation between operations in the category of graphs (the pullback and the pushout) and group theoretic operations between subgroups of free groups (the intersection and the join). After that, it is developed the theory of coverings of graphs. Finally, it is given an algorithm in order to decide whether a finitely generated subgroup of a free group is of finite index

Subgroups of direct products 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

On subdirect products of type FPnFP_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.Comment: Accepted in Math. Proc. Cambridge Philos. So

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 $\Sigma^1(G) = \emptyset = \Sigma^1(G, \mathbb{Q})$. For a group $H$ that is a finitely presented residually Droms RAAG we calculate $\Sigma^1(H)$ and $\Sigma^2(H)_{dis}$. In addition, we obtain a necessary condition for $[\chi]$ to belong to $\Sigma^n(H)$