Vol. 82, No. 1, 1979

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Metabelian representations of knot groups

Richard I. Hartley

Vol. 82 (1979), No. 1, 93–104

The question of determining which finite metabelian groups may be the homomorphic image of a given knot group G is considered in this paper. As a starting point, it is shown that a homomorphism of a knot group onto a metabelian group H such that [H : H] = n must factor through Zn???An, where An is the homology group of the n-fold cyclic covering space.

This is similar to a theorem of Burde [1 Satz 4|, and Reyner [5] has also proven a similar result, showing in effect that such a homomorphism must factor through Z An. Now, An can be given the structure of a module over the ring Ztof L-polynomials, and the problem of determining the metabelian factor groups of G can be reduced to determining the factor modules of An.

Mathematical Subject Classification 2000
Primary: 57M25
Received: 30 August 1978
Published: 1 May 1979
Richard I. Hartley