Free quotients of congruence subgroups of Bianchi groups
type: Item
instance of: scholarly article
title: Free quotients of congruence subgroups of Bianchi groups
zbMATH Open document ID: 0734.20021
DOI: 10.1007/BF01444703
published in: Mathematische Annalen
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164945
review text: 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, noncycli…
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 4214256
zbMATH Keywords: Dedekind ring of arithmetic type
zbMATH Keywords: subgroup of finite index
zbMATH Keywords: ring of integers
zbMATH Keywords: imaginary quadratic number field
zbMATH Keywords: coordinate ring
zbMATH Keywords: affine curve
zbMATH Keywords: projective curve
zbMATH Keywords: finite field
zbMATH Keywords: congruence subgroups
zbMATH Keywords: elementary matrices
zbMATH Keywords: principal congruence subgroup
zbMATH Keywords: Bianchi groups
zbMATH Keywords: free quotient
zbMATH Keywords: noncongruence cracks
zbMATH Keywords: standard subgroups
author: A. W. Mason
reviewed by: A. W. Mason
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214256