Combinatorial Methods: Free Groups, Polynomials, and Free Algebras
Author: Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
Published by Springer New York
ISBN: 978-1-4419-2344-8
DOI: 10.1007/978-0-387-21724-6

Table of Contents:

Introduction
Classical Techniques of Combinatorial Group Theory
Test Elements
Other Special Elements
Automorphic Orbits
The Jacobian Conjecture
The Cancellation Conjecture
Nagata’s Problem
The Embedding Problem
Coordinate Polynomials
Test Elements of Polynomial and Free Associative Algebras
Schreier Varieties of Algebras
Rank Theorems and Primitive Elements
Generalized Primitive Elements
Free Leibniz Algebras

Includes bibliographical references (pages 283-305) and indexes
Preface -- Introduction -- I. Groups: Introduction. Classical Techniques. Test Elements. Other Special Elements. Automorphic Orbits -- II. Polynomial Algebras: Introduction. The Jacobian Conjecture. The Cancellation Conjecture. Nagata's Problem. The Embedding Problem. Coordinate Polynomials. Test Polynomials -- III. Free Nielsen-Schreier Algebras: Introduction. Schreier Varieties of Algebras. Rank Theorems and Primitive Elements. Generalized Primitive Elements. Free Leibniz Algebras -- References -- Notations -- Author Index -- Subject Index
The main purpose of this book is to show how ideas from combinatorial group theory have spread to two other areas of mathematics: the theory of Lie algebras and affine algebraic geometry. Some of these ideas, in turn, came to combinatorial group theory from low-dimensional topology in the beginning of the 20th Century. This book is divided into three fairly independent parts. Part I provides a brief exposition of several classical techniques in combinatorial group theory, namely, methods of Nielsen, Whitehead, and Tietze. Part II contains the main focus of the book. Here the authors show how the aforementioned techniques of combinatorial group theory found their way into affine algebraic geometry, a fascinating area of mathematics that studies polynomials and polynomial mappings. Part III illustrates how ideas from combinatorial group theory contributed to the theory of free algebras. The focus here is on Schreier varieties of algebras (a variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras)
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