WRITING COMMUTATORS OF COMMUTATORS AS PRODUCTS OF CUBES IN GROUPS

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(gamma) of an element gamma of a group G, we mean the least natural number k such that gamma is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = FC infinity is the wreath product of a free group F by the infinite cyclic group, we show that every element of W '' is a product of at most three cubes in W.