Commutators and Squares in Free Nilpotent Groups

Abstract EN

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 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 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.