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ISSN 1088-6826(e) ISSN 0002-9939(p)

Non-normal, standard subgroups of the Bianchi groups

Author(s): A. W. Mason; 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 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 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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