Abstract: *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 which we call adorable groups. By definition
the derived series of these groups stabilizes at finite stage. We prove
some basic results on these groups and give a large class of examples.
We conjecture that fundamental groups of most nonpositively curved manifolds
are not adorable. Also we conjecture that if the successive quotients of
the derived series of a finitely presented torsion free group are finite
then the group is adorable. A particular case of the second conjecture
contains a part (half) of the virtual Betti number conjecture.*