A characterization of cohomological dimension for a big class of groups

It was shown by Cornick and Kropholler (1998) in [5] that if a group G is in hF then the finitistic dimension of ZG, , is equal to the supremum of the projective dimensions of the ZG-modules M such that pdZHM<∞ for all finite subgroups H of G. Here we show that a locally hF-group G is of type Φ if and only if . A group G is of type Φ if for any ZG-module M, pdZGM<∞ if and only if pdZHM<∞ for every finite subgroup H of G.Type Φ was proposed in Talelli (2007) [18] as an algebraic characterization for those groups G that admit a finite dimensional model for .We also deduce a number of corollaries for locally hF-groups.