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

Abstract: 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.