A criterion for detecting trivial elements of Burnside groups

TL;DR

This paper introduces a criterion to identify trivial elements in free Burnside groups of large odd exponent, aiding the study of their automorphisms without relying on their known infiniteness.

Contribution

It provides a new necessary and sufficient condition for triviality of elements in Burnside groups, independent of their known properties.

Findings

Criterion applies without prior knowledge of Burnside groups

Enables analysis of outer automorphisms of Burnside groups

Includes an analogue for hyperbolic group quotients

Abstract

In this article we give a sufficient and necessary condition to determine wether or not an element of the free group induces a non-trivial element of the free Burnside group of sufficiently large odd exponent. This criterion can be stated without any knowledge about Burnside groups, in particular about the proof of its infiniteness. Therefore it provides a useful tool that we will use later to study outer automorphisms of Burnside groups. We also state an analogue result for periodic quotients of torsion-free hyperbolic groups.