Brown's Criterion and classifying spaces for families

Abstract

Let $G$ be a group and $\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\mathcal{F}$, and for all $H\in \mathcal{F}$ the fixed point subspace $X^H$ is contractible. The group $G$ is of type $\mathcal{F}\text{-}\mathrm{F}_{n}$ if it admits a model for $E_\mathcal{F} G$ with $n$-skeleton with compact orbit space. The main result of the article provides is a characterization of $\mathcal{F}\text{-}\mathrm{F}_{n}$ analogue to Brown's criterion for $\mathrm{FP}_n$. As applications we provide criteria for this type of finiteness properties with respect to families to be preserved by finite extensions, a result that contrast with examples of Leary and Nucinkis. We also recover L\"uck's characterization of property $\underline{\mathrm{F}}_n$ in terms of the finiteness properties of the Weyl groups.