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

Grunewald, Fritz; Schwermer, Joachim

doi:10.1090/S0002-9947-1993-1020042-6

Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $3$-space
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by Fritz Grunewald and Joachim Schwermer

Trans. Amer. Math. Soc. 335 (1993), 47-78

DOI: https://doi.org/10.1090/S0002-9947-1993-1020042-6

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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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