Achiral words

TL;DR

This paper investigates the property of achirality in words within free groups, providing examples and linking the concept to Whitehead's algorithm, with a focus on Engel words.

Contribution

It introduces classes of achiral words and demonstrates how Whitehead's algorithm can determine achirality for Engel words.

Findings

Identifies classes of achiral words in free groups.

Shows Whitehead's algorithm can be used to check achirality.

Addresses the question about Engel words' achirality.

Abstract

A word $w$ in a free group is {\em achiral} if for every group $G,$ $G_w=G_{w^{-1}},$ where $G_w$ is the image of the word map $w$ on $G.$ We will give few classes of examples of achiral words. Cocke and Ho asked whether Engel words are achiral or not. We will prove that it is enough to apply Whitehead's algorithm to check the same.