The Burnside problem for odd exponents

TL;DR

This paper proves that free Burnside groups with an odd exponent of at least 557 are infinite, using advanced small cancellation theory and introducing a new concept called certification sequences.

Contribution

It establishes the infiniteness of free Burnside groups for a new lower bound of the odd exponent, improving previous results with novel proof techniques.

Findings

Free Burnside groups $B(m,n)$ are infinite for $m extgreater 1$ and odd $n extgreater 557$

Introduces the concept of certification sequences in small cancellation theory

Provides the best known lower bound for the exponent in Burnside groups

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.