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


Author: Youn W. Lee
Journal: Trans. Amer. Math. Soc. 290 (1985), 735-745
MSC: Primary 57S30; Secondary 20H15
DOI: https://doi.org/10.1090/S0002-9947-1985-0792824-2
MathSciNet review: 792824
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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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DOI: https://doi.org/10.1090/S0002-9947-1985-0792824-2
Article copyright: © Copyright 1985 American Mathematical Society

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