On normal torsion-free finite index subgroups of polyhedral groups
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).
polyhedron of finite volume in hyperbolic 3-space
scientific article; zbMATH DE number 4105886
scientific article