Palindromic words in simple groups

TL;DR

This paper proves that all simple groups have a finite generating set with palindromic width 1, while non-abelian finite simple groups can have larger palindromic width, and also explores properties of just-infinite groups.

Contribution

It establishes the existence of generating sets with minimal palindromic width in simple groups and introduces new examples of groups with finite palindromic width but infinite commutator width.

Findings

Every simple group has a generating set with palindromic width 1.

Non-abelian finite simple groups can have palindromic width greater than 1.

Every just-infinite group has finite palindromic width with respect to some finite generating set.

Abstract

A palindrome is a word that reads the same left-to-right as right-to-left. We show that every simple group has a finite generating set $X$, such that every element of it can be written as a palindrome in the letters of $X$. Moreover, every simple group has palindromic width $pw(G,X)=1$, where $X$ only differs by at most one Nielsen-transformation from any given generating set. On the contrary, we prove that all non-abelian finite simple groups $G$ also have a generating set $S$ with $pw(G,S)>1$. As a by-product of our work we also obtain that every just-infinite group has finite palindromic width with respect to a finite generating set. This provides first examples of groups with finite palindromic width but infinite commutator width.