A note on finiteness properties of vertex stabilisers

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 ${\mathsf{F}\mathsf{P}}_n$ and not of type ${\mathsf{F}\mathsf{P}}_{n+1}$.