Combinatorial Algebra: Syntax and Semantics

  • Mark V. Sapir

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Mark V. Sapir
    Pages 1-69
  3. Mark V. Sapir
    Pages 71-84
  4. Mark V. Sapir
    Pages 85-159
  5. Mark V. Sapir
    Pages 161-196
  6. Mark V. Sapir
    Pages 197-330
  7. Back Matter
    Pages 331-355

About this book


Combinatorial Algebra: Syntax and Semantics provides a comprehensive account of many areas of combinatorial algebra. It contains self-contained proofs of  more than 20 fundamental results, both classical and modern. This includes Golod–Shafarevich and Olshanskii's solutions of Burnside problems, Shirshov's solution of Kurosh's problem for PI rings, Belov's solution of Specht's problem for varieties of rings, Grigorchuk's solution of Milnor's problem, Bass–Guivarc'h theorem about the growth of nilpotent groups, Kleiman's solution of Hanna Neumann's problem for varieties of groups, Adian's solution of von Neumann-Day's problem, Trahtman's solution of the road coloring problem of Adler, Goodwyn and Weiss. The book emphasize several ``universal" tools, such as trees, subshifts, uniformly recurrent words, diagrams and automata.


With over 350 exercises at various levels of difficulty and with hints for the more difficult problems, this book can be used as a textbook, and aims to reach a wide and diversified audience.  No prerequisites beyond standard courses in linear and abstract algebra are required. The broad appeal of this book extends to a variety of student levels: from advanced high-schoolers to undergraduates and graduate students, including those in search of a Ph.D. thesis who will benefit from the  “Further reading and open problems” sections at the end of Chapters 2 –5.


The book can be used in a classroom and for self-study, engagin

g anyone who wishes to learn and better understand this important area of mathematics.


Burnside-type problems Novikov–Adian theorem amenable groups combinatorics on words semigroups symbolic dynamics

Authors and affiliations

  • Mark V. Sapir
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Bibliographic information