On algebraic characterizations for finiteness of the dimension of EG

Certain algebraic invariants of the integral group ring ZG of a group G were introduced and investigated in relation to the problem of extending the Farrell-Tate cohomology, which is defined for the class of groups of finite virtual cohomological dimension. It turns out that the finiteness of these invariants of a group G implies the existence of a generalized Farrell-Tate cohomology for G which is computed via complete resolutions.In this article we present these algebraic invariants and their basic properties and discuss their relationship to the generalized Farrell-Tate cohomology. In addition we present the status of conjecture which claims that the finiteness of these invariants of a group G is equivalent to the existence of a finite dimensional model for EG, the classifying space for proper actions.

Keywords:

Farrell-Tate cohomology , virtual cohomological dimension , complete resolution , finitistic dimension of the integral group ring , classifying space for proper action