Solvable normal subgroups of 2-knot groups

Abstract

If $X$ is an orientable, strongly minimal $P{D}_4$-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 $⟨a,t|ta=a^{2}t⟩$, 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.

arXiv commentThe introduction has been extended and there is a new section (partly speculative) on 2-knot groups of geometric dimension 2