METABELIAN SL(N, ℂ) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS

Given a knot K with complement NK and an irreducible metabelian SL(n, ℂ) representation α : π1(NK)→SL(n, ℂ), we establish the inequality dim H1(NK; sl(n, ℂ)Adα)≥n−1. In the case of equality, we prove that α must have finite image and is conjugate to an SU(n) representation. In this case, we show α determines a smooth point ξα in the SL(n, ℂ) character variety, and we use a deformation argument to establish the existence of a smooth (n−1)-dimensional family of characters of irreducible SL(n, ℂ) representations near ξα and a corresponding subfamily of characters of irreducible SU(n) representations of real dimension n−1. Both families can be chosen so that ξα is the only metabelian character. Combining this with our previous existence results, we deduce the existence of large families of irreducible SU(n) and SL(n, ℂ) non-metabelian representation for knots K in homology 3-spheres Σ with non-trivial Alexander polynomial. We then relate the condition on twisted cohomology to a more accessible condition on untwisted cohomology of a certain metabelian branched cover of Σ branched along K.