Normal subgroups of prescribed order and zero level of subgroups of the Bianchi groups (Q5927592)

Let \({\mathcal O}_d\) be the ring of integers of the imaginary quadratic number field \(\mathbb{Q}(\sqrt{-d})\), where \(d\) is a square-free, positive integer. Let \(E_2({\mathcal O}_d)\) denote the subgroup of \(\text{SL}_2({\mathcal O}_d)\) generated by the elementary matrices. For each \({\mathcal O}_d\)-ideal \(\mathfrak q\) let \(E_2({\mathcal O}_d,{\mathfrak q})\) be the normal subgroup of \(E_2({\mathcal O}_d)\) generated by the \(\mathfrak q\)-elementary matrices. Let \(S\) be a subgroup of \(\text{SL}_2({\mathcal O}_d)\). The level \(l(S)\) is the largest ideal \({\mathfrak q}_0\) with \(E_2({\mathcal O}_d,{\mathfrak q}_0)\subset S\). The order \(o(S)\) is the \({\mathcal O}_d\)-ideal generated by \(x_{ij}\) and \(x_{ii}-x_{jj}\) for \(i\neq j\) where \((x_{ij})\in S\). Let \[ {\mathcal N}_0({\mathcal O}_d;{\mathfrak q})=\{N\triangleleft\text{SL}_2({\mathcal O}_d)\mid o(N)={\mathfrak q},\;l(N)=\{0\}\}. \] Then for all but finitely many \(d\), \(\text{card }{\mathcal N}_0({\mathcal O}_d;{\mathfrak q})=2^{\aleph_0}\) for nonzero \(\mathfrak q\). This answers a question of A. Lubotzky.