The free Burnside group \(B(m,n)\) is the \(m\)-generated group defined by all relations of the form \(x^n=1\). Despite the simplicity of the definition, obtaining a structural information about the free Burnside groups is known to be a difficult problem. The primary question of this sort is whether \(B(m,n)\) is finite for given \(m, n \ge 2\). Starting from fundamental results of Novikov and Adian, it became known that \(B(m,n)\) is infinite for all sufficiently large exponents \(n\). There are known several approaches to prove this result and to establish other properties of groups \(B(m,n)\) in the `infinite' case. However, even simpler ones are quite technical and require a large lower bound on the exponent \(n\) (as odd \(n \gt 10^{10}\) in Ol'shanskii's approach).

The aim of the talk is to present yet another approach to free Burnside groups of odd exponent \(n\) with \(m\ge2\) generators based on a version of iterated small cancellation theory. The approach works for a `moderate' bound \(n \gt 2000\). In the introductory part, I make a brief survey of results around Burnside groups and give an informal introduction to the small cancellation theory.