Non-normal, standard subgroups of the Bianchi groups

Mason, A.; Odoni, R.

doi:10.1090/S0002-9939-96-03310-2

Non-normal, standard subgroups of the Bianchi groups
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by A. W. Mason and R. W. K. Odoni

Proc. Amer. Math. Soc. 124 (1996), 721-726

DOI: https://doi.org/10.1090/S0002-9939-96-03310-2

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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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