Writing Commutators of Commutators as Products of Cubes in Groups

Abstract

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(γ) of an element γ of a group G, we mean the least natural number k such that γ is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = F≀C ∞ 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.