Solvable normal subgroups of 2-knot groups

J.A.Hillman

Abstract

If the fundamental group of an orientable, strongly minimal PD4-complex has one end then it has no nontrivial locally-finite normal subgroup. Hence if a 2-knot group is virtually solvable then either it has two ends or is the group of Fox's Example 10, or is torsion-free and polycyclic of Hirsch length 4. If the centre of a 2-knot group \(G\) is nontrivial then either \(G\) has two ends, or \(G\) has one end and the centre is torsion-free, or \(G\) has infinitely many ends and the centre is finite. The Hirsch–Plotkin radical of any 2-knot group is nilpotent.

Keywords: centre, Hirsch-Plotkin radical, solvable, 2-knot.

AMS Subject Classification: Primary 57Q45.