The derived series and virtual Betti numbers
Siddhartha Gadgil
Abstract
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.
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