Groups with many finitary cohomology functors
  For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is a dichotomy: this set of natural numbers is either finite or cofinite. Moreover any finite or cofinite set is shown to arise in suitable examples. A survey of recent literature on this topic, especially the work of Martin Hamilton, is included. 
P. H. Kropholler
math.GR