  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.
