The derived series and virtual Betti numbers

الأصل · EN

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 Gⁱ/Gⁱ⁺¹ of the derived series Gⁱ of G are finite, then the derived series stabilises. The proof involves basic facts regarding finite group actions on homology spheres.