Finitary Group Cohomology and Eilenberg-Mac Lane Spaces
  We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n.   In this paper, we show that if G is a group in Kropholler's class LHF with cohomology almost everywhere finitary, then G has an Eilenberg--Mac Lane space K(G,1) which is dominated by a CW-complex with finitely many n-cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G.   We also remark that the converse holds for any group G. 
Martin Hamilton