Brown's criterion and classifying spaces for families

Abstract

Let G be a group and F be a family of subgroups closed under conjugation and subgroups. A model for the classifying space EFG is a G-CW-complex X such that every isotropy group belongs to F, and for all H∈F the fixed point subspace XH is contractible. The group G is of type F-Fn if it admits a model for EFG with n-skeleton with compact orbit space. The main result of the article provides is a characterization of F-Fn analogue to Brown's criterion for FPn. 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 Lück's characterization of property F_n in terms of the finiteness properties of the Weyl groups.