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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $3$-space
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by Fritz Grunewald and Joachim Schwermer PDF
Trans. Amer. Math. Soc. 335 (1993), 47-78 Request permission

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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 335 (1993), 47-78
  • MSC: Primary 11F06; Secondary 20H25, 22E40, 57N10
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1020042-6
  • MathSciNet review: 1020042