Abelian Rank of Normal Torsion-Free Finite Index Subgroups of Polyhedral Groups

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.