Free quotients of congruence subgroups of Bianchi groups
Let A be a Dedekind ring of arithmetic type. \textit{H. Bass}, \textit{J. Milnor} and \textit{J.-P. Serre} [Publ. Math., Inst. Hautes Etud. Sci. 33, 59-137 (1967; Zbl 0174.052)] have shown that every subgroup of finite index in \(SL_ n(A)\), where \(n\geq 3\), is ``essentially'' a congruence subgroup. \textit{L. Vaserstein} [Mat. Sb., Nov. Ser. 89, 313-322 (1972; Zbl 0359.20027)] and \textit{B. Liehl} [J. Reine Angew. Math. 323, 153-171 (1981; Zbl 0447.20035)] have proved that this is also true for \(SL_ 2(A)\) provided A has infinitely many units. It is known that if A has only finitely many units then (i) \(A={\mathbb{Z}}\), the ring of rational integers, (ii) \(A={\mathfrak O}(={\mathfrak O}_ d)\), the ring of integers of the imaginary quadratic number field \({\mathbb{Q}}(\sqrt{-d})\), where d is a square-free positive integer, or (iii) \(A={\mathfrak C}={\mathfrak C}(C,P,k)\), the coordinate ring of the affine curve obtained by removing a closed point P from a projective curve over a finite field k. \textit{J.-P. Serre} [Ann. Math., II. Ser. 92, 489-527 (1970; Zbl 0239.20063)], using a global approach involving the theory of profinite groups, has proved that for these latter cases \(SL_ 2(A)\) has ``many more'' subgroups of finite index than congruence subgroups. Let \(E_ 2(A)\) be the subgroup of \(SL_ 2(A)\) generated by elementary matrices. For each A-ideal q, let \(NE_ 2(A,q)\) be the normal subgroup of \(SL_ 2(A)\) generated by the q-elementary matrices and let \(SL_ 2(A,q)\) be the principal congruence subgroup of \(SL_ 2(A)\) (of level q). The main result of this paper is the following. Theorem: Let \(P=\{p: p\) prime and \(p\equiv 3(mod 4)\}\). There exists a finite set X with the following properties. Let \(d\not\in X\cup P\). Then, for all nonzero \({\mathfrak O}\)-ideals q, there exists \(N\triangleleft SL_ 2({\mathfrak O})\) such that (i) \(NE_ 2({\mathfrak O},q)\leq N\leq SL_ 2({\mathfrak O},q),\) (ii) \(SL_ 2({\mathfrak O},q)/N\) is free, noncyclic. (The groups \(SL_ 2({\mathfrak O})\) (or \(PSL_ 2({\mathfrak O}))\) are usually called the Bianchi groups.) The proof is based on the fundamental paper of \textit{R. Zimmer} [Invent. Math. 19, 73-81 (1973; Zbl 0254.10019)] together with recent joint work of the author with Odoni and Stothers. We also prove some partial results for the case where \(d\in P\). For \(d\not\in X\cup P\) this result extends earlier results of Cohn, Frohman and Fine on \(E_ 2({\mathfrak O})\). Zimmer, Grunewald and Schwermer have proved that every \(SL_ 2({\mathfrak O})\) has a subgroup of finite index with a free quotient. However their results say little about elementary matrices contained in the kernels of such quotients. The author has previously proved results similar to the above for the cases \(A={\mathbb{Z}}\) and \(A={\mathfrak C}.\) The author considers two applications of the results. For \(d\not\in X\cup P\) the above theorem is used to extend earlier results of Lubotzky on the existence of noncongruence cracks for \(SL_ 2({\mathfrak O})\). In addition the approach here is simpler than that of Lubotzky in that it avoids recourse to the theory of profinite groups. Earlier results of the author are used in a similar way to extend Lubotzky's corresponding results for \(SL_ 2({\mathbb{Z}})\) and (all) \(SL_ 2({\mathfrak C}).\) A subgroup S of \(GL_ n(R)\), where R is any commutative ring and \(n\geq 2\), is said to be standard if and only if \(E_ n(R,q_ 0)\leq S\), where \(E_ n(R,q_ 0)\) is the normal subgroup of \(E_ n(R)\) generated by the \(q_ 0\)-elementary matrices and \(q_ 0\) is the R-ideal generated by \(x_{ij}\), \(x_{ii}-x_{jj}\) (i\(\neq j)\), for all \((x_{ij})\in S\). Vaserstein has proved that S is standard if and only if S is normalized by \(E_ n(R)\), provided \(n\geq 3\). From the above theorem if follows that, when \(d\not\in X\cup P\), \(GL_ 2({\mathfrak O})\) has uncountable many standard subgroups which are not normalized by \(SL_ 2({\mathfrak O})\).
scientific article; zbMATH DE number 4214256
scientific article