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The Burnside problem for odd exponents

Agatha Atkarskaya, Eliyahu Rips, Katrin Tent

TL;DR

It is shown that the free Burnside groups B(n,m) are infinite for n ≥ 557 and m ≥ 2 and the best known lower bound in the Burnside problem for odd exponents from 665 to 557 is decreased.

Abstract

We show that the free Burnside groups $B(m,n)$ are infinite for $m\geq 2$ and odd $n\geq 557$, the best currently known lower bound for the exponent. The proof uses iterated small cancellation theory where the induction is based on the nesting depth of relators. The main instrument at every step is a new concept of a certification sequence.

The Burnside problem for odd exponents

TL;DR

It is shown that the free Burnside groups B(n,m) are infinite for n ≥ 557 and m ≥ 2 and the best known lower bound in the Burnside problem for odd exponents from 665 to 557 is decreased.

Abstract

We show that the free Burnside groups are infinite for and odd , the best currently known lower bound for the exponent. The proof uses iterated small cancellation theory where the induction is based on the nesting depth of relators. The main instrument at every step is a new concept of a certification sequence.
Paper Structure (22 sections, 111 theorems, 108 equations)

This paper contains 22 sections, 111 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. The set-up
  3. The list of induction hypotheses
  4. The induction
  5. Induction basis
  6. Cyclically canonical words
  7. Sets of relators $\mathrm{Rel}_r$ and their common parts
  8. Turns of rank $r$
  9. Inverse turns
  10. Influence of turns on other maximal occurrences
  11. Commuting turns of rank $r$
  12. $\lambda$-semicanonical forms of rank $r$
  13. Defining the canonical form of rank $r$
  14. Determining winner sides
  15. Rank $\mathbf{r = 1}$.
  16. ...and 7 more sections

Key Result

Theorem 2.1

The free Burnside group $B(m, n)$ is infinite for $m \geqslant 2$ and odd exponents $n \geqslant 557$.

Theorems & Definitions (263)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Definition 3.1: Fractional powers and $\Lambda_{i}$-measure
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 3.5
  • Definition 3.6
  • Corollary 3.7
  • ...and 253 more