A note on finiteness properties of vertex stabilisers

TL;DR

This paper establishes a criterion linking the finiteness properties of vertex stabilisers in G-CW-complexes to those of the entire group, extending previous results to higher dimensions and revealing diverse group classes.

Contribution

It generalizes Haglund--Wise's result from trees to higher-dimensional complexes, providing new insights into the finiteness properties of groups and their stabilisers.

Findings

Proves a criterion relating group and stabiliser finiteness properties.

Shows existence of uncountably many groups with specific finiteness properties.

Extends finiteness property results to higher-dimensional G-CW-complexes.

Abstract

We prove a criterion for the geometric and algebraic finiteness properties of vertex stabilisers of $G$-CW-complexes, given the finiteness properties of the group $G$ and of the stabilisers of positive dimensional cells. This generalises a result of Haglund--Wise for groups acting on trees to higher dimensions. As an application, for $n\ge 2$, we deduce the existence of uncountably many quasi-isometry classes of one-ended groups that are of type $\mathsf{FP}_n$ and not of type $\mathsf{FP}_{n+1}$.