Avoiding Square-Free Words on Free Groups

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.