Abtract: A discrete group G has periodic cohomology if there is an element in a cohomology group, cup product with which induces an isomorphism in cohomology after certain dimension. Adem and Smith showed that this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It was conjectured by Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this talk I will use the implied condition of jump cohomology to verify the conjecture for certain classes of groups. I will also show several finiteness properties for groups that act on finite dimensional complexes with certain cohomological conditions.