Powers of a product of commutators as products of squares

Abstract

We prove that for any odd integer N and any integer n>0, the Nth power of a product of n commutators in a nonabelian free group of countable infiniterank can be expressed as a product of squares of 2n+1 elements and, for all such odd N and integersn, there are commutators for which the number 2n+1 of squares is the minimum number such that the Nth power of its product can be written as a product of squares. This generalizes a recent result of Akhavan-Malayeri. Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.