Combinatorial Methods

Free Groups, Polynomials, and Free Algebras

  • Alexander A. Mikhalev
  • Vladimir Shpilrain
  • Jie-Tai Yu

Part of the CMS Books in Mathematics book series (CMSBM)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Introduction

    1. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 1-3
  3. Groups

    1. Front Matter
      Pages 5-8
    2. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 9-19
    3. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 20-34
    4. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 35-44
    5. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 45-64
  4. Polynomial Algebras

    1. Front Matter
      Pages 65-70
    2. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 71-79
    3. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 80-91
    4. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 92-107
    5. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 108-128
    6. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 129-168
    7. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 169-181
  5. Free Nielsen-Schreier Algebras

    1. Front Matter
      Pages 183-189
    2. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 190-212
    3. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 213-243
    4. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 244-269
    5. Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 270-282
  6. Back Matter
    Pages 283-315

About this book

Introduction

 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).

Keywords

Dimension Group theory Lie Topology algebra algebraic geometry geometry mapping mathematics theorem

Authors and affiliations

  • Alexander A. Mikhalev
    • 1
  • Vladimir Shpilrain
    • 2
  • Jie-Tai Yu
    • 3
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Department of MathematicsThe City College of New YorkNew YorkUSA
  3. 3.Department of MathematicsThe University of Hong KongHong KongChina

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-21724-6
  • Copyright Information Springer-Verlag New York 2004
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-2344-8
  • Online ISBN 978-0-387-21724-6
  • Series Print ISSN 1613-5237
  • About this book