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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $ 3$-space


Authors: Fritz Grunewald and Joachim Schwermer
Journal: Trans. Amer. Math. Soc. 335 (1993), 47-78
MSC: Primary 11F06; Secondary 20H25, 22E40, 57N10
MathSciNet review: 1020042
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Abstract: Let $ \mathcal{O}$ be the ring of integers in an imaginary quadratic number-field. The group $ {\text{PSL}}_2(\mathcal{O})$ acts discontinuously on hyperbolic $ 3$-space $ H$. If $ \Gamma \leq {\text{PSL}}_2(\mathcal{O})$ is a torsionfree subgroup of finite index then the manifold $ \Gamma \backslash H$ can be compactified to a manifold $ {M_\Gamma }$ so that the inclusion $ \Gamma \backslash H \leq {M_\Gamma }$ is a homotopy equivalence. $ {M_\Gamma }$ is a compact with boundary. The boundary being a union of finitely many $ 2$-tori. This paper contains a computer-aided study of subgroups of low index in $ {\text{PSL}}_2(\mathcal{O})$ for various $ \mathcal{O}$. The explicit description of these subgroups leads to a study of the homeomorphism types of the $ {M_\Gamma }$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1020042-6
PII: S 0002-9947(1993)1020042-6
Article copyright: © Copyright 1993 American Mathematical Society