Short description:

 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).

Long description:

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.

From the reviews:

"This book is devoted to a combinatorial theory of three types of objects: (1) free groups, (2) polynomial algebras and free associative algebras, (3) free algebras of the so-called Nielsen-Schreier varieties of algebras. It considers problems related mainly to the groups of automorphisms of these objects... The authors have done a lot of work to show that the same problems and the same ideas are the moving forces of the three theories. The book contains a good background on the classical results (most of them without proof) and a detailed exposition of the recent results. A large portion of the exposition is devoted to topics in which the authors have made their own contribution." -- MATHEMATICAL REVIEWS

"The book consists of three parts: groups, polynomial algebras and free Nielsen-Schreier algebras. ? The book contains very interesting material to which the authors have made a valuable contribution. The book includes many open and very important problems. ? The exposition of the material is made with care. So the book could be recommended for students even as a textbook." (Vyacheslav A. Artamonov, Zentralblatt MATH, Vol. 1039 (8), 2004)

Table of Contents:

I Groups.- 1 Classical Techniques of Combinatorial Group Theory.- 2 Test Elements.- 3 Other Special Elements.- 4 Automorphic Orbits.- II Polynomial Algebras.- 5 The Jacobian Conjecture.- 6 The Cancellation Conjecture.- 7 Nagata?s Problem.- 8 The Embedding Problem.- 9 Coordinate Polynomials.- 10 Test Polynomials.- III Free Nielsen-Schreier Algebras.- 11 Schreier Varieties of Algebras.- 12 Rank Theorems and Primitive Elements.- 13 Generalized Primitive Elements.- 14 Free Leibniz Algebras.- References.- Notation Index.- Author Index.