Ebook: Combinatorial Methods: Free Groups, Polynomials, and Free Algebras

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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 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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 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).

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-3
Front Matter....Pages 5-8
Classical Techniques of Combinatorial Group Theory....Pages 9-19
Test Elements....Pages 20-34
Other Special Elements....Pages 35-44
Automorphic Orbits....Pages 45-64
Front Matter....Pages 65-70
The Jacobian Conjecture....Pages 71-79
The Cancellation Conjecture....Pages 80-91
Nagata’s Problem....Pages 92-107
The Embedding Problem....Pages 108-128
Coordinate Polynomials....Pages 129-168
Test Elements of Polynomial and Free Associative Algebras....Pages 169-181
Front Matter....Pages 183-189
Schreier Varieties of Algebras....Pages 190-212
Rank Theorems and Primitive Elements....Pages 213-243
Generalized Primitive Elements....Pages 244-269
Free Leibniz Algebras....Pages 270-282
Back Matter....Pages 283-315

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 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).

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-3
Front Matter....Pages 5-8
Classical Techniques of Combinatorial Group Theory....Pages 9-19
Test Elements....Pages 20-34
Other Special Elements....Pages 35-44
Automorphic Orbits....Pages 45-64
Front Matter....Pages 65-70
The Jacobian Conjecture....Pages 71-79
The Cancellation Conjecture....Pages 80-91
Nagata’s Problem....Pages 92-107
The Embedding Problem....Pages 108-128
Coordinate Polynomials....Pages 129-168
Test Elements of Polynomial and Free Associative Algebras....Pages 169-181
Front Matter....Pages 183-189
Schreier Varieties of Algebras....Pages 190-212
Rank Theorems and Primitive Elements....Pages 213-243
Generalized Primitive Elements....Pages 244-269
Free Leibniz Algebras....Pages 270-282
Back Matter....Pages 283-315
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