Primitive elements in the free metabelian group of rank \(3\)
type: Item
instance of: scholarly article
title: Primitive elements in the free metabelian group of rank \(3\)
zbMATH Open document ID: 0938.20027
author: Martin J. Evans
published in: Journal of Algebra
publication date: 2000-04-03T00:00:00Z
review text: Let \(R_m=\mathbb{Z}[x_1^{\pm 1},\ldots,x_m^{\pm 1}]\) denote the Laurent polynomial ring in indeterminates \(x_1,\ldots,x_m\) with coefficients in \(\mathbb{Z}\). We let \(\text{SL}_n(R_m)\) denote the special linear group and \(E_n(R_m)\) denote the subgroup of \(\text{SL}_n(R_m)\) generated by the elementary matrices in \(\text{SL}_n(R_m)\). The well known existence theorem of \textit{S. Bachmuth} and \textit{H. Y. Mochizuki} [Am. J. Math. 104, 1181-1189 (1982; Zbl 0513.20038)] states that \(\text{SL}_2(R_m)\not=E_2(R_m)\) for \(m\geq 2\) (the case \(m=1\) is still open). This is an exceptional property because a famous result of \textit{A. A. Suslin} [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 235-252 (1977; Zbl 0354.13009)] asserts that \(\text{SL}_n(R_m)=E_n(R_m)\) for \(n\geq 3\) and every \(m\). Let \(F_n\) be the free group and \(M_n\) be the free metabelian group of rank \(n\). It is well known from the results of Bachmuth and Mochizuki (and of the reviewer) that every automorphism of \(M_n\) is induced by some automorphism of \(F_n\) for \(n\not=3\). The case \(n=3\) is exceptional. \textit{S. Bachmuth} and \textit{H. Y. Mochizuki} [J. Algebra 55, 106-115 (1978; Zbl 0401.20033)] proved that \(\Aut M_3\) is infinitely generated. Moreover, the reviewer [\textit{V. A. Roman'kov}, Mat. Sb. 182, No. 7, 1074-1085 (1991; Zbl 0743.20035), English transl. in Math. USSR, Sb. 73, No. 2, 445-455 (1992)] proved that there exists a primitive element of \(M_3\) that is not the image of a primitive element of \(F_3\) under the natural map from \(F_3\) to \(M_3\). No specific elements of this type have appeared in the literature and the main purpose of the paper is to exhibit such one. Thus, the author gives examples of matrices in \(\text{SL}_2(R_2)\setminus E_2(R_2)\), and examples of specific primitive elements in \(M_3\) which are not induced by primitive elements in \(F_3\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20H25
zbMATH DE Number: 1384470
zbMATH Keywords: free metabelian groups
zbMATH Keywords: free groups
zbMATH Keywords: primitive elements
zbMATH Keywords: Laurent polynomial rings
zbMATH Keywords: special linear groups
zbMATH Keywords: elementary subgroups
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
full work available at URL: jabr.1998.7817
OpenAlex ID: W2020592047
cites work: Induced Automorphisms of Free Groups and Free Metabelian Groups
cites work: IA-automorphisms of the free metabelian group of rank 3
cites work: E 2 � SL 2 for Most Laurent Polynomial Rings
cites work: Aut(F) →Aut(F/F � � ) is Surjective for Free Group F of Rank 4
cites work: IA automorphisms of free and free metabelian groups
cites work: Presentations of Groups Involving More Generators than are Necessary
cites work: Presentations of the Free Metabelian Group of Rank 2
cites work: Q4866223
cites work: Q5513066
cites work: Q3744373
cites work: Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen
cites work: Varieties of groups
cites work: Q3980658
cites work: ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS
DOI: 10.1006/JABR.1998.7817
description: scientific article; zbMATH DE number 1384470