On the groups \(\mathrm{SL}_ 2(\mathbb{Z}[x])\) and \(\mathrm{SL}_ 2(k[x,y])\)
type: Item
instance of: scholarly article
title: On the groups \(\mathrm{SL}_ 2(\mathbb{Z}[x])\) and \(\mathrm{SL}_ 2(k[x,y])\)
zbMATH Open document ID: 0805.20042
DOI: 10.1007/BF02773676
author: Jens Mennicke
author: L. N. Vaserstein
author: Fritz Grunewald
published in: Israel Journal of Mathematics
publication date: 1995-02-08T00:00:00Z
review text: Let \(R = \mathbb Z[x]\) or \(k[x,y]\), where \(\mathbb Z\) is the ring of rational integers and \(k\) is a finite field, and let \(U_ 2(R)\) be the subgroup of \(\text{SL}_ 2(R)\) generated by the unipotent matrices. The authors prove that \(\text{SL}_ 2(R)/U_ 2(R)\) has a free quotient of any finite rank. This result is very much a two-dimensional anomaly since Suslin has proved that, when \(n \geq 3\), \(\text{SL}_ n(R) = E_ n(R)\), where \(E_ n(R)\) is the subgroup generated by the elementary matrices. Using a technique involving symplectic matrices it is shown that, if \(S\) is a quotient of \(R\), then the corresponding natural map \(\phi: \text{SL}_ 2(R) \to \text{SL}_ 2(S)\) is ``almost surjective''. The proof of the main result follows from looking at \(\text{Im }\phi\) for various \(S\). When \(R = \mathbb Z[x]\), \(S\) is the ring of integers of some imaginary quadratic number fields. When \(R = k[x,y]\), \(S\) is a ring of integers in a hyperelliptic function field over \(k\). The paper also contains results on the exact stable range of various polynomial rings, together with structure theorems for the group \(U_ 2(R)/\widehat{E}_ 2(R)\), when \(\widehat{E}_ 2(R)\) is the normal subgroup of \(\text{SL}_ 2(R)\) generated by \(E_ 2(R)\).
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 19B14
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 590937
zbMATH Keywords: unipotent matrices
zbMATH Keywords: free quotient of any finite rank
zbMATH Keywords: elementary matrices
zbMATH Keywords: symplectic matrices
zbMATH Keywords: imaginary quadratic number fields
zbMATH Keywords: hyperelliptic function field
zbMATH Keywords: stable range
zbMATH Keywords: normal subgroup
reviewed by: A. W. Mason
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590937