Avoiding Square-Free Words on Free Groups

TL;DR

This paper investigates the structure of infinite square-free words in free groups, constructing examples that avoid certain subwords, analyzing their frequency and growth, and establishing bounds on avoided patterns.

Contribution

It introduces new constructions of infinite Dean words avoiding specific reduced words and analyzes their properties, including frequency and growth rate.

Findings

Infinite Dean words avoiding six reduced words of length 3 exist.

Minimal letter frequency in Dean words is at least 8/59.

Growth rate of these words is approximately 1.45818.

Abstract

We consider sets of factors that can be avoided in square-free words on two-generator free groups. The elements of the group are presented in terms of 0,1,2,3 such that 0 and 2 (resp.,1 and 3) are inverses of each other so that 02, 20, 13 and 31 do not occur in a reduced word. A Dean word is a reduced word that does not contain occurrences of $uu$ for any nonempty $u$. Dean showed in 1965 that there exist infinite square-free reduced words. We show that if $w$ is a Dean word of length at least 59 then there are at most six reduced words of length 3 avoided by $w$. We construct an infinite Dean word avoiding six reduced words of length~3. We also construct infinite Dean words with low critical exponent and avoiding fewer reduced words of length 3. Finally, we show that the minimal frequency of a letter in a Dean word is $8/59$ and the growth rate is close to 1.45818.