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

Abstract In this paper, we study the relation between the finiteness properties of subgroups of direct products of 2-dimensional coherent right-angled Artin groups with their structure and the decidability of algorithmic problems. More precisely, we show that a finitely presented subgroup S of the direct product of 2-dimensional coherent RAAGs is virtually a nilpotent extension of a direct product. Moreover, if S is of type FP, then S is commensurable to a kernel of a character. We use these results to show that the multiple conjugacy problem and the membership problem are decidable for finitely presented subgroups of direct products of 2-dimensional coherent RAAGs. This work generalizes the results of Bridson, Howie, Miller and Short for free groups.