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Groups

  • Mark V. Sapir
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we introduce a new tool – van Kampen diagrams (although these are sibling of diagrams introduced in Section  1.7.4). We start with defining van Kampen diagrams and explaining basic methods of using them: the bands, the Swiss cheese method and the small cancelation theory. Then we explain Golod’s solution of the unbounded Burnside problem and a road map of Olshanskii’s proof of the Novikov–Adian theorem solving the bounded Burnside problem: there exists an infinite finitely generated group satisfying the identity x n  = 1 for all odd n ≥ 665. The theorem was one of the main achievements in group theory of the 20th century. Its initial proof occupied more than 300 pages and was extremely complicated. Olshanskii managed to find a much simpler proof (for much bigger n). Our road map explains the main ideas and “points of interest” of Olshanskii’s proof.

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

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