Abstract: * This is a continuation of an earlier preprint under
the
same title. These papers grew out of an attempt to find a suitable
finite
sheeted covering of an aspherical 3-manifold so that the cover either
has infinite or trivial first homology group. With this motivation we
defined a new class of groups. These groups are in some sense
eventually
perfect. Here we prove results giving several classes of examples of
groups which do (not) belong to this class. Also we state two
conjectures.
A direct application of one of the conjectures to the virtual Betti
number
conjecture is mentioned. The other conjecture says that
most nonpositively curved Riemannian manifold have fundamental groups
which are not eventually perfect. For completeness, here we reproduce
parts of the previous paper.*