Solvable normal subgroups of 2-knot groups

TL;DR

This paper investigates the structure of 2-knot groups, establishing conditions under which their normal subgroups are trivial or have specific algebraic properties, thereby advancing understanding of their algebraic and topological characteristics.

Contribution

It classifies the normal subgroups of 2-knot groups, especially regarding their solvability, ends, and the structure of their centers, providing new insights into their algebraic structure.

Findings

2-knot groups with one end have no nontrivial locally-finite normal subgroups

If a 2-knot group is virtually solvable, it is either two-ended, isomorphic to a specific presentation, or torsion-free and polycyclic of Hirsch length 4

The center of a 2-knot group is torsion-free when the group has one end, and finite when it has infinitely many ends

Abstract

If $X$ is an orientable, strongly minimal $PD_4$-complex and $\pi_1(X)$ has one end then it has no nontrivial locally-finite normal subgroup. Hence if $\pi$ is a 2-knot group then (a) if $\pi$ is virtually solvable then either $\pi$ has two ends or $\pi\cong\Phi$, with presentation $\langle{a,t}|ta=a^2t\rangle$, or $\pi$ is torsion-free and polycyclic of Hirsch length 4; (b) either $\pi$ has two ends, or $\pi$ has one end and the centre $\zeta\pi$ is torsion-free, or $\pi$ has infinitely many ends and $\zeta\pi$ is finite; and (c) the Hirsch-Plotkin radical $\sqrt\pi$ is nilpotent.