# Normal subgroups of prescribed order and zero level of subgroups of the Bianchi groups

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## Abstract.

Abstract. Let *S* be a subgroup of *SL* _{ n }(*R*), where *R* is a commutative ring with identity and \(n \geqq 3\). The *order* of *S, o*(*S*), is the *R*-ideal generated by \(x_{ij},\ x_{ii} - x_{jj}\ (i \neq j)\), where \((x_{ij}) \in S\). Let *E* _{ n }(*R*) be the subgroup of *SL* _{ n }(*R*) generated by the elementary matrices. The *level* of *S, l(S)*, is the *largest R*-ideal \(\frak {q}\) with the property that *S* contains all the \(\frak {q}\)-elementary matrices and all conjugates of these by elements of *E* _{ n }(*R*). It is clear that \(l(S) \leqq o(S)\). Vaserstein has proved that, for all *R* and for all \(n \geqq 3\), the subgroup *S* is normalized by *E* _{ n }(*R*) if and only if *l*(*S*) = *o*(*S*).¶Let *A* be an arithmetic Dedekind domain of characteristic zero with only finitely many units. It is known that \(A = \Bbb {Z}\) or \(A = {\cal O}_d\), the ring of integers in the imaginary quadratic field \(\Bbb {Q}(\sqrt {- d})\), where *d* is a square-free positive integer. It has been shown that, for all non-zero \(\Bbb {Z}\)-ideals \(\frak {q}\), there exist uncountably many normal subgroups of \(SL_2(\Bbb {Z})\) with order \(\frak {q}\) and level zero. In this paper we extend this result to all but finitely many of the *Bianchi groups* \(SL_2({\cal O}_d)\). This answers a question of A. Lubotzky.

## Keywords

Positive Integer Normal Subgroup Commutative Ring Characteristic Zero Level Zero## Preview

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