English

Solvable normal subgroups of 2-knot groups

Geometric Topology 2021-02-24 v3

Abstract

If XX is an orientable, strongly minimal PD4PD_4-complex and π1(X)\pi_1(X) has one end then it has no nontrivial locally-finite normal subgroup. Hence if π\pi is a 2-knot group then (a) if π\pi is virtually solvable then either π\pi has two ends or πΦ\pi\cong\Phi, with presentation a,tta=a2t\langle{a,t}|ta=a^2t\rangle, or π\pi is torsion-free and polycyclic of Hirsch length 4; (b) either π\pi has two ends, or π\pi has one end and the centre ζπ\zeta\pi is torsion-free, or π\pi has infinitely many ends and ζπ\zeta\pi is finite; and (c) the Hirsch-Plotkin radical π\sqrt\pi is nilpotent.

Keywords

Cite

@article{arxiv.1610.06629,
  title  = {Solvable normal subgroups of 2-knot groups},
  author = {J. A. Hillman},
  journal= {arXiv preprint arXiv:1610.06629},
  year   = {2021}
}

Comments

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

R2 v1 2026-06-22T16:27:19.140Z