Combinatorial methods : free groups, polynomials, and free algebras / Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu

Notes bibliogr. Index

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.
For many years the theories of these three types of objects have been developed without having much influence on each other. Combinatorial group theory is the oldest one and was created at the beginning of the 20th century to address the needs of topology. The combinatorial theory of polynomials was inspired mainly by algebraic geometry and the need to perform computations with concrete commutative algebras; combinatorial theory of free associative algebras often mimicked combinatorial commutative algebra. Finally, the theory of free Nielsen-Schreier algebras generalized combinatorial theory of free Lie algebras which was influenced by the theory of free groups. Of course, there have been some more or less occasional interactions between these three theories. The real and systematic cooperation between the theories of free groups, polynomial and free associative algebras and free Lie and free Nielsen-Schreier algebras started much later, maybe in the 1970's and 1980's.
The book treats combinatorial theory of free groups, polynomial algebras and free Nielsen-Schreier algebras from the same point of view. 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.
I believe that the algebraic community will find the book interesting and useful. The text is suitable both for beginners and experts. Each of the parts of the book (free groups, polynomial algebras, free Nielsen-Schreier algebras) could serve as a course on combinatorial theory of the corresponding objects, but the lecturer would need to select some chapters and sections of the corresponding part (and perhaps to extend the exposition of the course with some other text). Another possibility is to use the book for a course on topics similar to those included in each part of the book or to include chapters of the book in a graduate course on combinatorial algebra. It will also serve as a good reference book. ... (MathSciNet)