Cohomological nonvanishing for modules over discrete groups Original Research Article

Let Γ be a discrete group of finite virtual cohomological dimension, and let V be a f.g. Z-torsion free Γ module. In this situation, the Farrell cohomology of Γ with coefficients in V is defined and we prove in this paper cohomological non-vanishing results for these groups similar to those existing for finite groups and Tate cohomology, i.e. that we either have that these groups vanish for all dimensions or there are infinitely many nontrivial. The proof is based on a geometric approach to these groups, the study of Euler characteristics, minimal resolutions, and the notion of exponent.