Square-free words as products of commutators

Abstract

Elements of the commutator subgroup of a free group 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 wg which can be presented by f(g) distinct Wicks forms, where f(g) > g!. Moreover we may choose these words wg to be square-free.