Subgroups of direct products of limit groups over Droms RAAGs

TL;DR

This paper extends known subgroup structure results from limit groups over free groups to those over Droms RAAGs, demonstrating their subgroup properties, conjugacy problem solvability, and subgroup separability.

Contribution

It generalizes subgroup structure theorems to limit groups over Droms RAAGs and proves solvability of the conjugacy problem and subgroup separability for these groups.

Findings

Subgroups of certain limit groups over Droms RAAGs are virtually direct products.

The conjugacy problem is solvable for finitely presented residually Droms RAAG groups.

Finitely presentable subgroups of these groups are separable.

Abstract

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.