Universität zu Köln

Boden, Hans U. and Friedl, Stefan (2014). METABELIAN SL(N, C) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS. Q. J. Math., 65 (3). S. 817 - 841. OXFORD: OXFORD UNIV PRESS. ISSN 1464-3847

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Abstract

Given a knot K with complement N-K and an irreducible metabelian SL(n, C) representation alpha : pi(1)(N-K) -> SL(n, C), we establish the inequality dim H-1(N-K; sl(n, C)(Ad alpha)) >= n - 1. In the case of equality, we prove that alpha must have finite image and is conjugate to an SU(n) representation. In this case, we show alpha determines a smooth point xi(alpha) 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 xi(alpha) and a corresponding subfamily of characters of irreducible SU(n) representations of real dimension n - 1. Both families can be chosen so that xi(alpha) 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 Sigma 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 (Sigma) over cap (phi) of Sigma branched along K.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Boden, Hans U. UNSPECIFIED UNSPECIFIED UNSPECIFIED Friedl, Stefan UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-429778
DOI:	10.1093/qmath/hat047
Journal or Publication Title:	Q. J. Math.
Volume:	65
Number:	3
Page Range:	S. 817 - 841
Date:	2014
Publisher:	OXFORD UNIV PRESS
Place of Publication:	OXFORD
ISSN:	1464-3847
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language TWISTED ALEXANDER POLYNOMIALS; REDUCIBLE REPRESENTATIONS; 3-MANIFOLD GROUPS; VARIETIES; HOMOLOGY; GROWTH Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/42977