A note on higher coherence of graphs of groups

TL;DR

This paper investigates the property of higher coherence in graphs of groups, showing that under certain conditions, the entire structure inherits this property, with applications to specific classes like right-angled Artin groups.

Contribution

It establishes conditions under which graphs of groups with certain vertex and edge groups are higher coherent, extending understanding of group coherence properties.

Findings

Graphs of groups with n-coherent vertex groups and virtually poly-cyclic edges are n-coherent.

Certain right-angled Artin groups are shown to be n-coherent.

Provides new insights into the structure and coherence of complex group constructions.

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.