Commutators and Squares in Free Nilpotent Groups

In a free group no nontrivial commutator is a square. And in the free 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 class 3 freely generated by x1, x2. We prove that in F2,3 = 〈x1, x2〉, it is possible to write certain commutators as a square. We denote by Sq(γ) the minimal number of squares which is required to write γ as a product of squares in a group G. And we define Sq(G) = sup{Sq(γ); γ ∈ G′}. We discuss when the square length of a given commutator of F2,3 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. Keyword and phrases: commutator, square length, free nilpotent group AMS subject Classification 2010: 20F12;20F99