Published online by Cambridge University Press: 06 July 2010
In C. T. C. Wall conjectured that if a countable group G of finite virtual cohomological dimension, vcd G < ∞, has periodic Farrell cohomology then G acts freely and properly on ℝn × Sm for some n and m. Obviously, if a group G acts freely and properly on some ℝn × Sm then G is countable since ℝn × Sm is a separable metric space. The Farrell cohomology generalizes the Tate cohomology theory for finite groups to the class of groups G with vcd G < ∞ (see for instance Chapter X of). Wall's conjecture was proved by Johnson in some cases and Connolly and Prassidis in general.
In Prassidis showed that there are groups of infinite vcd which act freely and properly on some ℝn × Sm. In particular, it follows from results of Prassidis and Talelli that if a countable group G has periodic cohomology after 1-step then G acts freely and properly on some ℝn × Sm.
A group G is said to have periodic cohomology after κ-steps if there is a positive integer q such that the functors Hi(G;) and Hi+q(G;) are naturally equivalent for all i > κ (cf.).