Commutators and Squares in Free Nilpotent Groups

In a free group no nontrivial commutator is a square. And in thefree group F2=F(x1,x2) freely generated by x1,x2 the commutator [x1,x2] is never the product of two squares in F2, although it is always the product of three squares. Let F2,3=〈x1,x2〉 be a free nilpotent group of rank 2 and class3 freely generated by x1,x2. We prove that in F2,3=〈x1,x2〉, it is possibleto write certain commutators as a square. We denote by Sq(γ) the minimalnumber of squares which is required to write γ as a product of squares in group G. And we define Sq(G)=sup{Sq(γ);γ∈G′}.We discuss the question of when the square length of a given commutator ofF2,3 is equal to 1 or 2 or 3. The precise formulas for expressing any commutator of F2,3 as the minimal number of squares are given. Finally as an application of these results we prove that Sq(F′2,3)=3