The derived series and virtual Betti numbers

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.