On normal torsion-free finite index subgroups of polyhedral groups
type: Item
instance of: scholarly article
title: On normal torsion-free finite index subgroups of polyhedral groups
zbMATH Open document ID: 0675.57018
DOI: 10.1016/0166-8641(89)90004-7
author: Youn W. Lee
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let P be a convex polyhedron of finite volume in hyperbolic 3-space all of whose dihedral angles are of the form \(\pi\) /n, where n is an integer. Then the subgroup of index 2 of orientation-preserving elements in the group generated by the reflection in the faces of the polyhedron is called the polyhedral group associated to P. In the present paper it is shown that the abelianization of a normal torsion-free finite index subgroup in a polyhedral group is never isomorphic to the integers \({\mathbb{Z}}\); especially, such a normal subgroup is not the group of a knot in the 3-sphere (however, the group of the figure-8-knot is a subgroup of index 12, necessarily non-normal, in a tetrahedral group; this seems to be the only known example of such an inclusion).
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 4105886
zbMATH Keywords: knot group
zbMATH Keywords: polyhedron of finite volume in hyperbolic 3-space
zbMATH Keywords: group generated by the reflection
zbMATH Keywords: polyhedral group
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90004-7
OpenAlex ID: W1968376705
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description: scientific article; zbMATH DE number 4105886