The Burnside groups and small cancellation theory

Summary

Abstract

In a pair of recent articles, the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. More specifically, these articles prove that the free Burnside groups of exponent n ≥ 1260 are infinite groups which have a decidable word problem. The structure of the finite subgroups of the free Burnside groups is calculated from the automorphism groups of the general relators used in the presentation. The present article gives a brief introduction to the methods and techniques involved in the proof. Many of the ideas originate with the recent work of A. Yu. Ol'shanskii and S. V. Ivanov. Some familiarity with their work on the Burnside problem will be assumed.

Introduction

Ninety-five years ago William Burnside asked whether every finitely generated group of finite exponent must be finite. Since then, significant progress has been made in the study of the free Burnside groups, much of it coming in recent years. In this first section the recent history of their study will be briefly reviewed so that the results announced below can be viewed in their proper context. The second section introduces many of the concepts from which the general theory is constructed. The third and the fourth sections will describe the results of general small cancellation theory and the results on the Burnside groups in greater detail.