Commutators as products of squares

Abstract

It is shown that if G G is the free group of rank 2 freely generated by x x and y y , then x y x − 1 y − 1 xy{x^{ - 1}}{y^{ - 1}} is never the product of two squares in G G , although it is always the product of three squares in G G . It is also shown that if G G is the free group of rank n n freely generated by x 1 , x 2 , ⋯ , x n {x_1},{x_2}, \cdots ,{x_n} , then x 1 2 x 2 2 ⋯ x n 2 x_1^2x_2^2 \cdots x_n^2 is never the product of fewer than n n squares in G G .