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Solvable normal subgroups of 2-knot groups

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

E-mail Address: jonathan.hillman@sydney.edu.au

https://doi.org/10.1142/S0218216517500663Cited by:0 (Source: Crossref)

Abstract

If $X$ is an orientable, strongly minimal $P D_{4}$ -complex and $π_{1} (X)$ has one end, then it has no nontrivial locally finite normal subgroup. Hence, if $π$ is a 2-knot group, then (a) if $π$ is virtually solvable, then either $π$ has two ends or $π ≅ Φ$ , with presentation $〈 a, t | t a = a^{2} t 〉$ , or $π$ is torsion-free and polycyclic of Hirsch length 4 (b) either $π$ has two ends, or $π$ has one end and the center $ζ π$ is torsion-free, or $π$ has infinitely many ends and $ζ π$ is finite, and (c) the Hirsch–Plotkin radical $\sqrt{π}$ is nilpotent.

Keywords:

AMSC: 57Q45

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