Vol. 105, No. 2, 1983

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Lifting group homomorphisms

Richard I. Hartley

Vol. 105 (1983), No. 2, 311–320

If a knot K has Alexander polynomial different from 1, then its knot group, G maps onto some metacyclic group, Zr???Zp. We show that in that case, it also has a homomorphism onto a split extension of a free abelian group of rank p1 by Zr???Zp, and hence also onto a split extension of a direct sum of p1 cyclic groups of order s by the metacyclic group. In many cases, (such as if s is coprime with p), this group can be specified exactly. Otherwise there are a finite number of possibilities. A special case is Perko’s result that a homomorphism of a knot group onto S3 = Z2???Z3 lifts to S4 = Z2???Z3???(Z2 Z2).

As an application we obtain information about the derived series of G.

In a final section it is shown how to associate a rational polynomial invariant to every metacyclic representation.

Mathematical Subject Classification 2000
Primary: 57M25
Received: 24 November 1980
Published: 1 April 1983
Richard I. Hartley