  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.
