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 Luck's characterization of property $\underline{\mathrm{F}}_n$ in terms of the finiteness properties of the Weyl groups.

中文翻译：

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布朗标准和家庭空间分类

设 $G$ 是一个群，$\mathcal{F}$ 是一个在共轭和子群下封闭的子群族。分类空间 $E_{\mathcal{F}} G$ 的模型是 $G$-CW-复数 $X$，使得每个各向同性群都属于 $\mathcal{F}$，并且对于所有的 $H\在 \mathcal{F}$ 中，不动点子空间 $X^H$ 是可收缩的。群 $G$ 的类型为 $\mathcal{F}\text{-}\mathrm{F}_{n}$ 如果它承认 $E_\mathcal{F} G$ 的模型和 $n$-skeleton具有紧凑的轨道空间。本文提供的主要结果是 $\mathcal{F}\text{-}\mathrm{F}_{n}$ 的表征，类似于 $\mathrm{FP}_n$ 的布朗准则。作为应用程序，我们提供了关于要通过有限扩展保留的族的此类有限性属性的标准，该结果与 Leary 和 Nucinkis 的示例形成对比。我们也恢复运气'