Abelian rank of normal torsion-free finite index subgroups of polyhedral groups

Lee, Youn W.

doi:10.1090/S0002-9947-1985-0792824-2

Abelian rank of normal torsion-free finite index subgroups of polyhedral groups
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Trans. Amer. Math. Soc. 290 (1985), 735-745 Request permission

Abstract:

Suppose that $P$ is a convex polyhedron in the hyperbolic $3$-space with finite volume and $P$ has integer $( > 1)$ submultiples of $\pi$ as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group $G$ associated to $P$ is one, then $P$ has exactly one ideal vertex of type $(2,2,2,2)$ and $G$ has an index two subgroup which does not contain any one of the four standard generators of the stabilizer of the ideal vertex.

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