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Let CU($\gamma$)be the minimal number of cubes required
to express an element $\gamma$ 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($\gamma$) for certain
elements of the derived subgroup of $F$. If $W=F\imath C_ {\infty}$ is
the wreath product of $F$ by the infinite cyclic group, we also
show that every element of $W^\prime$ is a product of at most one
commutator and three cubes in $W$.
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