A note on higher coherence of graphs of groups

Abstract

For $n\in \mathbb{N}$, a group is called $n$-coherent if every subgroup of type $\mathsf{F}_n$ is of type $\mathsf{F}_{n+1}$. For $n\ge 1$, we observe that graphs of groups with $n$-coherent vertex groups and virtually poly-cyclic edge groups are $n$-coherent. We deduce the $n$-coherence of certain right-angled Artin groups.