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

TL;DR

This paper generalizes Baumslag and Roseblade's result to a broader class of graphs of groups with free abelian vertex groups, showing that finitely presented subgroups are virtually structured as extensions of direct products.

Contribution

It extends the virtual structural description of finitely presented subgroups from free groups to cyclic subgroup separable graphs of groups with free abelian vertices, including RAAGs and tubular groups.

Findings

Finitely presented subgroups are virtually $H$-by-(free abelian).

Decidability of multiple conjugacy and membership problems for certain RAAGs.

Applicability to 2-dimensional coherent right-angled Artin groups and tubular groups.

Abstract

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.