Combinatorial Methods

by Mikhalev, Alexander A.; Shpilrain, Vladimir; Yu, Jie-Tai

• ISBN13: 9780387405629
• ISBN10: 0387405623
• Format: Hardcover
• Copyright: 2003-12-01
• Publisher: Springer Verlag

Summary

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 at 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).

Author Biography

Alexander A. Mikhalev is Professor in the Department of Mechanics at Mathematics at Moscow State University.

Table of Contents

Preface	vii
Introduction	1	(4)
I Groups	5	(60)
Introduction	7	(2)
1 Classical Techniques of Combinatorial Group Theory	9	(11)
1.1 Nielsen's Method	9	(2)
1.2 Whitehead's Method	11	(2)
1.3 Tietze's Method	13	(1)
1.4 Free Differential Calculus	14	(6)
1.4.1 Two applications	17	(3)
2 Test Elements	20	(15)
2.1 Nielsen's Commutator Test	21	(4)
2.2 Recognizing Test Elements	25	(5)
2.3 Test Sets	30	(3)
2.4 The "Double Jacobian" Matrix	33	(2)
3 Other Special Elements	35	(10)
3.1 Δ-Primitive Elements	35	(6)
3.2 Almost Primitive and Generic Elements	41	(4)
4 Automorphic Orbits	45	(20)
4.1 Finite Orbits	46	(2)
4.2 Abridged Orbits	48	(11)
4.2.1 Counting primitive elements	48	(5)
4.2.2 Complexity of Whitehead's algorithm	53	(6)
4.3 Endomorphisms that Preserve Automorphic Orbits	59	(6)
II Polynomial Algebras	65	(118)
Introduction	67	(4)
5 The Jacobian Conjecture	71	(9)
5.1 Polynomial Retracts and the Jacobian Conjecture	72	(3)
5.2 Retracts of K[x, y]	75	(1)
5.3 Applications to the Jacobian Conjecture	76	(4)
6 The Cancellation Conjecture	80	(12)
6.1 Equivalence and Stable Equivalence	83	(5)
6.2 Around Danielewski's Example	88	(4)
7 Nagata's Problem	92	(16)
7.1 Gröbner Bases and Face Functions	95	(6)
7.2 Lifting the Nagata Automorphism	101	(7)
8 The Embedding Problem	108	(21)
8.1 Embeddings of Curves in the Plane	109	(9)
8.1.1 Elementary automorphisms and peak reduction	111	(2)
8.1.2 A classification of parametric curves	113	(2)
8.1.3 Embeddings of curves in the plane	115	(3)
8.2 Embeddings of Hypersurfaces in Affine Space	118	(9)
8.2.1 Invariants of isomorphic varieties	122	(2)
8.2.2 Inequivalent isomorphic varieties	124	(3)
8.3 The Embedding Conjecture for Free Associative Algebras	127	(2)
9 Coordinate Polynomials	129	(40)
9.1 Coordinates of Two-Variable Polynomial Algebras	130	(8)
9.1.1 Proofs of main results	132	(4)
9.1.2 Algorithm for detecting coordinate polynomials	136	(1)
9.1.3 Relation to the Jacobian Conjecture	137	(1)
9.2 Coordinates in Free Associative Algebra of Rank Two	138	(6)
9.2.1 Fox derivatives in a free associative algebra	141	(1)
9.2.2 Proofs and algorithms	142	(2)
9.3 Density of Coordinates in Polynomial Algebras	144	(3)
9.4 Coordinates in K [z] [x, y]	147	(22)
9.4.1 Preliminaries	148	(3)
9.4.2 Characterization of tame and wild coordinates in K [z] [x, y]	151	(7)
9.4.3 New class of wild automorphisms	158	(11)
10 Test Polynomials	169	(14)
10.1 Test Polynomials in Polynomial Algebras	171	(3)
10.2 Test Elements in Free Associative Algebras	174	(6)
10.3 Open Problems	180	(3)
III Free Nielsen-Schreier Algebras	183	(123)
Introduction	185	(5)
11 Schreier Varieties of Algebras	190	(23)
11.1 Main Types of Nielsen-Schreier Algebras	191	(5)
11.2 Schreier Techniques	196	(5)
11.3 Free Differential Calculus	201	(7)
11.3.1 Free differential calculus and Schreier varieties	201	(6)
11.3.2 Ranks of subalgebras	207	(1)
11.4 Stable Equivalence	208	(2)
11.5 The Rank of an Endomorphism	210	(3)
12 Rank Theorems and Primitive Elements	213	(31)
12.1 Basic Properties of Partial Derivatives	214	(3)
12.2 Homogeneous Admissible Elements	217	(2)
12.3 Elimination of Variables	219	(6)
12.4 Rank Theorems	225	(4)
12.5 Primitive Elements	229	(8)
12.5.1 Primitive elements in free nonassociative algebras	230	(1)
12.5.2 Primitive elements in free Lie superalgebras	230	(5)
12.5.3 The Freiheitssatz and primitive elements	235	(2)
12.6 Primitive Elements and Endomorphisms	237	(4)
12.7 Inverse Images of Primitive Elements	241	(3)
13 Generalized Primitive Elements	244	(26)
13.1 Test Elements	247	(5)
13.2 Retracts	252	(2)
13.3 Endomorphisms and Automorphic Orbits	254	(2)
13.4 Almost Primitive and Test Elements	256	(6)
13.5 Δ-Primitive Elements of Free Lie Algebras	262	(2)
13.6 Generic Elements of Free Lie Algebras	264	(6)
14 Free Leibniz Algebras	270	(36)
14.1 Basic Notions	271	(2)
14.2 Differential Separability of Subalgebras	273	(2)
14.3 Residual Finiteness of Free Leibniz Algebras	275	(4)
14.4 Automorphisms of Free Leibniz Algebras	279	(4)
14.4.1 Simple automorphisms	279	(2)
14.4.2 Recognizing tame automorphisms	281	(2)
References	283	(23)
Notation Index	306	(2)
Author Index	308	(5)
Subject Index	313