Combinatorial methods. Free groups, polynomials, and free algebras
type: Item
instance of: scholarly article
title: Combinatorial methods. Free groups, polynomials, and free algebras
zbMATH Open document ID: 1039.16024
author: A. A. Mikhalev
author: Vladimir Shpilrain
author: Jie-Tai Yu
published in: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC
publication date: 2004-02-17T00:00:00Z
review text: The book consists of three parts: groups, polynomial algebras and free Nielsen-Schreier algebras. In each of these related classes the authors consider problems of orbits of automorphism groups, test elements, tame and wild automorphisms and some other topics. The first part starts with the Nielsen-Schreier theorem, Whitehead's method of determining whether two elements of a free group belong to one automorphic orbit of the automorphism group of a free group and Tietze's method of isomorphism of groups with given presentations. Also free differential Fox calculus in integral group rings of a free group is exposed. An element \(g\) of a group \(G\) is a test element if for any endomorphism \(\phi\) of the group \(G\) the equality \(\phi(g)=g\) implies that \(\phi\) is an automorphism. Properties of test elements in free groups are considered in chapter 2 using the free differential calculus. Let \(J\) be a right ideal of the integral group ring \(\mathbb Z F\) of a free group \(F\). An element \(u\in F\) is \(J\)-primitive if the Fox derivatives of \(u\) generate \(J\) as a right ideal. Properties of \(\Delta\)-primitive elements for the augmentation ideal \(\Delta\) in \(\mathbb Z F\) are discussed in chapter 3. In chapter 4 the authors consider the following problems. Is an endomorphism of a free group an automorphism if it (i) preserves the set of primitive elements, (ii) preserves a nontrivial automorphism orbit? The second part is concerned with the Jacobian conjecture and with the following problems. (i) Is it the case that a retract of the polynomial algebra \(k[X_1,\dots,X_n]\) is isomorphic to a polynomial algebra? (ii) Is every automorphism of the polynomial algebra \(k[X_1,\ldots,X_n]\) tame? (iii) Is every automorphism of a polynomial algebra \(k[X_1,\dots,X_n]\) at least stably tame? Note that the second problem was negatively solved recently by \textit{U. U. Umirbaev} and \textit{I. P. Shestakov} [J. Am. Math. Soc. 17, No. 1, 197--227 (2004; Zbl 1056.14…
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 17A32
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 2043994
zbMATH Keywords: free group
zbMATH Keywords: varieties of algebras
zbMATH Keywords: automorphisms
zbMATH Keywords: polynomial algebras
zbMATH Keywords: free Nielsen-Schreier algebras
zbMATH Keywords: automorphism groups
zbMATH Keywords: test elements
zbMATH Keywords: tame automorphisms
zbMATH Keywords: wild automorphisms
zbMATH Keywords: integral group rings
zbMATH Keywords: free groups
zbMATH Keywords: augmentation ideal
zbMATH Keywords: free associative algebras
zbMATH Keywords: homogeneous varieties
zbMATH Keywords: variety of Lie algebras
zbMATH Keywords: Schreier varieties
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: Leibniz algebras
zbMATH Keywords: Gröbner bases
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2043994