Square-free words as products of commutators

Abstract

Elements of the commutator subgroup of a free group $\mathbb{F}$ can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus $g$, there is a sequence of words $w_g$ which can be presented by $f(g)$ distinct Wicks forms, where $f(g)>g!$. Moreover we may choose these words $w_g$ to be square-free.