On groups of type $\Phi$

Olympia Talelli


Abstract.

The classifying space \underbar{E}G for the family of finite subgroups
of G is defined up to G-homotopy as a G-CW-complex \underbar{E}G such that 
\underbar{E}G^H is contractible if H is finite and empty otherwise.

We define a group G to be of type \Phi if it has the property that for every
ZG-module M, proj.dim_ZG M < \infty iff proj.dim_ZH M < \infty for every
finite subgroup H of G.

We show that if N_G(H)/H is of type \Phi for every finite subgroup H of G
and dim |\Lambda(G)| < \infty, where |\Lambda(G)| is the G-simplicial complex
determined by the poset of non-trivial finite subgroupss of G, then G admits
a finite dimensional model for \underbar{E}G.

This generalizes a theorem of P. Kropholler and G. Mislin [6] and a theorem
of W. L\"uck [8].

We also propose the type \Phi as an algebraic characterization of those 
groups which admit a finite dimensional model for \underbar{E}G.