Writing certain commutators as products of cubes in free groups

Let Cu(γ) be the minimal number of cubes required to express an element γ of a free group F. We establish a method for showing that certain equations do not have solutions in free groups. Using it, we find Cu(γ) for certain elements of the derived subgroup of F. If W=F C∞ is the wreath product of F by the infinite cyclic group, we also show that every element of W′ is a product of at most one commutator and three cubes in W.