%0 Journal Article
%T The derived series and virtual Betti numbers
%A Siddhartha Gadgil
%J Mathematics
%D 2003
%I arXiv
%X The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group $G$ of a hyperbolic three-manifold satisfies a certain stability property. The stability property is the statement that if all the quotients $G^i/G^{i+1}$ of the derived series $G^i$ of $G$ are finite, then the derived series stabilises. The proof involves basic facts regarding finite group actions on homology spheres.
%U http://arxiv.org/abs/math/0306359v1