Vol. 238, No. 1, 2008

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Metabelian SL(n, ) representations of knot groups

Hans U. Boden and Stefan Friedl

Vol. 238 (2008), No. 1, 7–25

We give a classification of irreducible metabelian representations from a knot group into SL(n, ) and GL(n, ). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n, ) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2, ) representations of knot groups. Finally we deduce the existence of irreducible metabelian SL(n, ) representations of the knot group for any knot with nontrivial Alexander polynomial.

metabelian representation, knot group, Alexander polynomial, branched cover
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 20C15
Received: 29 March 2008
Accepted: 24 July 2008
Published: 1 November 2008
Hans U. Boden
Department of Mathematics
McMaster University
Hamilton, Ontario L8S 4K1
Stefan Friedl
Mathematics Institute
University of Warwick
Coventry CV4 7AL
United Kingdom