Knot groups whose bases are abelian
Any knot group may be obtained as an HNN extension with finitely presentable base and associated subgroups. Here n-knot groups which are HNN extensions of finitely generated abelian groups are studied. (All knot groups with abelian commutator subgroup have this structure, by a theorem of \textit{R. Bieri} and \textit{R. Strebel} [Comment. Math. Helv. 53, 258-278 (1978; Zbl 0373.20035)].) Theorem 3.2 characterizes such n-knot groups when \(n\geq 3\); the result is analogous to the characterization of metabelian n-knot groups given by \textit{J. Levine} and \textit{C. Weber} [Lect. Notes Math. 685, 243-273 (1978; Zbl 0404.57019)] and includes it as a special case. In Section 4 it is shown that when \(n=2\) either the commutator subgroup is finite cyclic or \({\mathbb{Z}}^ 3\) or the base is \({\mathbb{Z}}\oplus ({\mathbb{Z}}/\lambda {\mathbb{Z}})\) for some \(\lambda >0\). In all known examples of the latter type \(\lambda =1\). The final section completes the description of abelian commutator subgroups of n-knot groups (n\(\geq 3)\) of rank 1 or 2 begun by \textit{J. C. Hausmann} and \textit{M. Kervaire} [Enseign. Math., II. Sér. 24, 111-123 (1978; Zbl 0414.57012)].
HNN extensions of finitely generated abelian groups
abelian commutator subgroups of n-knot groups
scientific article; zbMATH DE number 3949315
scientific article