Canadian Mathematical Sodety Sodete mathematique du Canada

Editors-in-Chief Redacteurs-en-chef Jonathan Borwein Peter Borwein

Springer Science+Business Media, LLC CMS Books in Mathematics Ouvrages de mathematiques de la SMC

HERMANlKuCERAISIMSA Equations and Inequalities 2 ARNOLD Abelian Groups and Representations ofFinite Partially Ordered Sets 3 BORWEIN/LEWIS Convex Analysis and Nonlinear Optimization 4 LEVIN/LuBINSKY Orthogonal Polynomials for Exponential Weights 5 KANE Reflection Groups and Invariant Theory 6 PHILLIPS Two Millennia ofMathematics 7 DEUTSCH Best Approximation in Inner Product Spaces 8 FABIAN ET AL. Functional Analysis and Infmite-Dimensional Geometry 9 KRiZEKlLucAlSOMER 17 Lectures on Fermat Numbers 10 BORWEIN Computational Excursions in Analysis and Number Theory 11 REED/SALES (Editors) Recent Advances in Algorithms and Combinatorics 12 HERMANlKuCERAISIMSA Counting and Configurations 13 NAZARETH Differentiable Optimization and Equation Solving 14 PHILLIPS Interpolation and Approximation by Polynomials 15 BEN-IsRAErJGREVILLE Generalized Inverses, Second Edition 16 ZHAO Dynamical Systems in Population Biology 17 GÖPFERT ET AL. Variational Methods in Partially Ordered Spaces 18 AKIvIS/GOLDBERG Differential Geometry of Varieties with Degenerate Gauss Maps 19 MIKHALEv/SHPILRAINNu Combinatorial Methods Alexander A. Mikhalev Vladimir Shpilrain Jie-Tai Yu Combinatorial Methods Free Groups, Polynomials, and Free Algebras

, Springer Alexander A. Mikhalev Vladimir Shpilrain Department of Mechanies Department of Mathematics and Mathematics Tbe City College ofNew York Moscow State University New York, NY 10031 Moscow 119899 USA Russia [email protected] [email protected] Jie-Tai Yu Department of Mathematics The University ofHong Kong Pokfulam Road HongKong China [email protected] Editors-in-Chief Redacteurs-en-chef Jonathan Borwein Peter Borwein Centre for Experimental and Constructive Mathematics Department ofMathematics and Statistics Simon Fraser University Canada [email protected] Mathematics Subject Classification (2000): 14-xx, 14Rxx, 17Bxx, 20Fxx

Library ofCongress Cataloging-in-Publication Data Shpilrain, Vladimir, 19~ Combinatorial methods : free groups, polynomials, and free algebras / Vladimir Shpilrain, Alexander A. Mikhalev, Jie-Tai Yu. p. cm. - (CMS books in mathematics series ; v. 19) Includes bibliographical references and index.

1. Combinatorial group theory. 2. Lie algebras. 3. Polynomials. I. Mikhalev, Alexander A.,196S- n. Yu, Jie-Tai. III. Title. IV. CMS books in mathematics ; 19. QAI82.S.sS7 2003 S12'2--dc22 2003058951 ISBN 978-1-4419-2344-8 ISBN 978-0-387-21724-6 (eBook) DOI 10.1007/978-0-387-21724-6 Printed on acid-free paper.

© 2004 SpringerScience+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2004 Softcover reprint ofthe hardcover 1st edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (SpringerScience+Business Media New York), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

987 6 S 4 3 2 1 SPIN 10940891

Typesetting: Pages created by the authors using a Springer laTeX 2e macro package. www.springer-ny.com Ta aur parents Preface

This book is about three seemingly independent areas of mathematics: combinatorial group theory, the theory of Lie algebras and affine algebraic geometry. Indeed, for many years these areas were being developed fairly independently. Combinatorial group theory, the oldest of the three, was born in the beginning of the 20th century as a branch of low-dimensional topology. Very soon, it became an important area of mathematics with its own powerful techniques. In the 1950s, combinatorial group theory started to influence, rather substantially, the theory of Lie algebrasj thus combinatorial theory of Lie algebras was shaped, although the origins of the theory can be traced back to the 1930s. In the 1960s, B. Buchberger introduced what is now known as Gröbner bases. This marked the beginning of a new, "combinatorial", era in commu• tative algebra. It is not very likely that Buchberger was directly influenced by ideas from combinatorial group theory, but his famous algorithm bears resemblance to Nielsen's method, although in a more sophisticated form. More recently, in the 1990s, ideas from combinatorial group theory started to have a more direct and significant impact on the areas of commutative algebra and affine algebraic geometry through the work of several mathe• maticians including the authors of this book. It is remarkable that, also in the 1990s, ideas from algebraic geometry found their way into combinato• rial group theory through the work of G. Baumslag, A. G. Myasnikov, V. Remeslennikov, O. Kharlampovich and B. 1. Plotkin, thus creating a "two-way traffic" of ideas that allows algebraic geometry to shed light on group theory and vice versa. viii Preface

The main purpose of this book is to show how ideas from combinatorial group theory have spread to the other two areas, with the main focus on the area of commutative algebra where the infiuence of these ideas has been especially spectacular. We would like to emphasize that we only consider here purely combinato• rial methods and results; in partieular, we leave out important interactions of group theory with topology and geometry that were recently used with great success in solving several difficult problems about free groups and their automorphisms. We also leave out geometrie methods in affine al• gebraic geometry and concentrate on combinatorial ones, in particular on those that come from combinatorial group theory. The book is targeted at research mathematicians as weIl as graduate students with an interest in the general area of algebra.

Acknowledgments

It is a great pleasure for us to thank everyone who has directly or indirectly contributed to this book. Our special thanks go to E. I. Zelmanov for his everlasting support and encouragement. We are also gratenIl to V. A. Ar• tamonov, V. Drensky, A. van den Essen, K. Q. Feng, L. Makar-Limanov, A. V. Mikhalev, U. U. Umirbaev, and A. A. Zolotykh for numerous helpful comments and insightftIl discussions. A. Pentus has helped us greatly in preparing a camera-ready copy of the manuscript. We would also like to thank various institutions where all or some of us stayed or visited during the preparation of this book. First of all, the University of Hong Kong and its Institute of Mathematical Research brought together all three of us on several occasions. Our thanks also go personally to N. Mok, the Director of the Institute of Mathematieal Re• search, M. K. Siu, the Chairman of the Department of Mathematics, and K. Y. Chan, the former Chairman of the Department of Mathematies, for their enthusiastie support of the algebraic program over many years. We are also grateftIl to our two other horne institutions, Moscow State University and the City College of New York, for a stimulating research environment. Finally, we are indebted to the staff of Springer-Verlag, in partieular, to M. Spenser and Y.-S. Hwang, for their patience and help during the production process.

Moscow, Russia Alexander A. Mikhalev New York, NY, USA Vladimir Shpilrain Hong Kong Jie-Tai Yu Contents

Preface vii

Introduction 1

I Groups 5

Introduction 7

1 Classical Techniques of Combinatorial Group Theory 9 1.1 Nielsen's Method . . 9 1.2 Whitehead's Method . . . 11 1.3 Tietze's Method ...... 13 1.4 Free Differential Calculus . 14 1.4.1 Two applications 17

2 Test Elements 20 2.1 Nielsen's Commutator Test. 21 2.2 Recognizing Test Elements. 25 2.3 Test Sets ...... 30 2.4 The "Double Jacobian" Matrix 33

3 Other Special Elements 35 3.1 ß-Primitive Elements...... 35 3.2 Almost Primitive and Generic Elements 41 x Contents

4 Automorphic Orbits 45 4.1 Finite Orbits...... 46 4.2 Abridged Orbits ...... 48 4.2.1 Counting primitive elements 48 4.2.2 Complexity of Whitehead's algorithm . 53 4.3 Endomorphisms that Preserve Automorphic Orbits 59

11 Polynomial Algebras 65

Introduction 67

5 The Jacobian Conjecture 71 5.1 Polynomial Retracts and the Jacobian Conjecture 72 5.2 Retracts of K[x, y] ...... 75 5.3 Applications to the Jacobian Conjecture 76

6 The Cancellation Conjecture 80 6.1 Equivalence and Stable Equivalence 83 6.2 Around Danielewski's Example 88

7 Nagata's Problem 92 7.1 Gröbner Bases and Face Functions 95 7.2 Lifting the Nagata Automorphism. 101

8 The Embedding Problem 108 8.1 Embeddings of Curves in the Plane ...... 109 8.1.1 Elementary automorphisms and peak reduction 111 8.1.2 A classification of parametric curves . 113 8.1.3 Embeddings of curves in the plane .. 115 8.2 Embeddings of Hypersurfaces in Affine Space 118 8.2.1 Invariants of isomorphie varieties . . . 122 8.2.2 Inequivalent isomorphie varieties ... 124 8.3 The Embedding Conjecture for Free Associative Algebras 127

9 Coordinate Polynomials 129 9.1 Coordinates of Two-Variable Polynomial Algebras . 130 9.1.1 Proofs of main results ...... 132 9.1.2 Algorithm for detecting coordinate polynomials 136 9.1.3 Relation to the Jacobian Conjecture ...... 137 9.2 Coordinates in Free Associative Algebra of Rank Two . 138 9.2.1 Fox derivatives in a free associative algebra 141 9.2.2 Proofs and algorithms ...... 142 9.3 Density of Coordinates in Polynomial Algebras 144 9.4 Coordinates in K[z] [x, y] 147 9.4.1 Preliminaries ...... 148 Contents xi

9.4.2 Characterization of tame and wild coordinates in K[z] [x, y] ...... 151 9.4.3 New dass of wild automorphisms ...... 158

10 Test Polynomials 169 10.1 Test Polynomials in Polynomial Aigebras . 171 10.2 Test Elements in Free Associative Aigebras . 174 10.3 Open Problems ...... 180

III Free Nielsen-Schreier Algebras 183

Introduction 185

11 Schreier Varieties of Aigebras 190 11.1 Main Types of Nielsen-Schreier Aigebras 191 11.2 Schreier Techniques ...... 196 11.3 Free Differential Calculus ...... 201 11.3.1 Free differential calculus and Schreier varieties . 201 11.3.2 Ranks of subalgebras . . 207 11.4 Stable Equivalence ...... 208 11.5 The Rank of an Endomorphism 210

12 Rank Theorems and Primitive Elements 213 12.1 Basic Properties of Partial Derivatives 214 12.2 Homogeneous Admissible Elements 217 12.3 Elimination of Variables 219 12.4 Rank Theorems ...... 225 12.5 Primitive Elements ...... 229 12.5.1 Primitive elements in free nonassociative algebras 230 12.5.2 Primitive elements in free Lie superalgebras 230 12.5.3 The Freiheitssatz and primitive elements 235 12.6 Primitive Elements and Endomorphisms 237 12.7 Inverse Images of Primitive Elements ...... 241

13 Generalized Primitive Elements 244 13.1 Test Elements ...... 247 13.2 Retracts ...... 252 13.3 Endomorphisms and Automorphic Orbits. 254 13.4 Almost Primitive and Test Elements ... 256 13.5 ~-Primitive Elements of Free Lie Aigebras 262 13.6 Generic Elements of Free Lie Aigebras 264

14 Free Leibniz Aigebras 270 14.1 Basic Notions ...... 271 14.2 Differential Separability of Subalgebras 273 xii Contents

14.3 Residual Finiteness of Free Leibniz Aigebras 275 14.4 Automorphisms of Free Leibniz Aigebras 279 14.4.1 Simple automorphisms ...... 279 14.4.2 Recognizing tarne automorphisms 281

References 283

Notation Index 306

Author Index 308

Subject Index 313