Title: Topology of 3-manifolds and a class of groups ( math.GT/0209121 )

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.

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