A note on higher coherence of graphs of groups

TL;DR

The paper introduces and develops the framework of higher coherence, defining $(n,m)$-coherence and its homological analogue, and proves a graph-of-groups combination theorem that lifts $(n,m)$-coherence from vertex and edge groups to the whole group. It extends these ideas to homological finiteness properties over rings and connects them to geometric-dimension and $L^2$-invariants, yielding conditions under which groups are homologically $(n,m)$-coherent. A key application is to right-angled Artin groups: for finite flag complexes $L$ in a specially constructed class $\mathcal{T}_n$, the RAAG $A_L$ is $(n,\infty)$-coherent, with explicit glueing and cone-operations preserving coherence. The work also presents examples showing limits of coherence results (e.g., some $A_L$ are not $(1,2)$- or $(2,3)$-coherent but are $(4,\infty)$-coherent) and discusses open questions, notably the $(3,\infty)$-coherence case, highlighting the nuanced landscape of higher finiteness properties in geometric group theory.

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.

Theorems & Definitions (22)

• Definition 1.1
• Theorem 1.2: Karrass--Solitar KS70
• Theorem 1.3
• Definition 1.4
• Theorem 1.5: Droms Droms87
• Definition 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2: GL
• Theorem 2.3