Abelian normal subgroups of two-knot groups (Q1087190)

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.