  This paper 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 define a new class of groups.
These groups are in some sense eventually perfect. We prove results giving
several classes of examples of groups which do (not) belong to this class. Also
we prove some elementary results on these groups and state two conjectures. A
direct application of one of the conjectures to the virtual Betti number
conjecture of Thurston is mentioned.
