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

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.