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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Non-normal, standard subgroups
of the Bianchi groups


Authors: A. W. Mason and R. W. K. Odoni
Journal: Proc. Amer. Math. Soc. 124 (1996), 721-726
MSC (1991): Primary 20H10, 11F06; Secondary 11A25, 20E05
MathSciNet review: 1322935
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Abstract: Let $S$ be a subgroup of $SL_n(K)$, where $K$ is a Dedekind ring, and let $\mathbf{q}$ be the $K$-ideal generated by $x_{ij},x_{ii}-x_{jj}$ $(i\ne j)$, where $(x_{ij})\in S$. The subgroup $S$ is called standard iff $S$ contains the normal subgroup of $SL_n(K)$ generated by the $\mathbf{q}$-elementary matrices. It is known that, when $n\ge 3$, $S$ is standard iff $S$ is normal in $SL_n(K)$. It is also known that every standard subgroup of $SL_2(K)$ is normal in $SL_2(K)$ when $K$ is an arithmetic Dedekind domain with infinitely many units.

The ring of integers of an imaginary quadratic number field, $\mathcal{O}$, is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group $SL_2(\mathcal{O})$ has uncountably many non-normal, standard subgroups. This result is already known for related groups like $SL_2(\mathbb{Z})$.


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Additional Information

A. W. Mason
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland

R. W. K. Odoni
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: awm@maths.gla.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03310-2
PII: S 0002-9939(96)03310-2
Received by editor(s): September 25, 1994
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1996 American Mathematical Society