Abelian normal subgroups of two-knot groups
type: Item
instance of: scholarly article
title: Abelian normal subgroups of two-knot groups
zbMATH Open document ID: 0611.57015
DOI: 10.1007/BF02621906
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140044
review text: For \(n\neq 2\) it is known which abelian groups can appear as subgroups of the centre of an n-knot group. An earlier paper of the author [ibid. 56, 465-473 (1981; Zbl 0478.57016)] considered the case of \(n=2\), but a gap was later found in the proof of a key lemma. The present paper fills that gap, and considers the more general question of abelian normal subgroups of 2-knot groups. It is shown that if such a subgroup is torsion free and of rank \(r>1\), then the knot group is an orientable Poincaré duality group of formal dimension 4, and so \(r\leq 4.\) The cases \(r=3\), 4 are settled completely, whilst many examples for \(r=1\), 2 can be constructed by twist-spinning classical knots. However, as the author points out, there are examples of 2-knot groups with rank 1 abelian normal subgroups which cannot be realized by fibred knots. He goes on to show that any virtually solvable 2-knot group must be either virtually poly-\({\mathbb{Z}}\) or an example given by R. H. Fox which has commutator subgroup the dyadic rationals or admit no nontrivial torsion free abelian normal subgroup.
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 3988278
zbMATH Keywords: centre of an n-knot group
zbMATH Keywords: abelian normal subgroups of 2-knot groups
zbMATH Keywords: Poincaré duality group
zbMATH Keywords: fibred knots
zbMATH Keywords: virtually solvable 2-knot group
author: Jonathan A. Hillman
reviewed by: C. Kearton
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3988278