On the lower central quotients of the free groups of finite rank of a certain variety

Abstract

Let F n be a free group of rank n, and γ k ( F n ) the k-th term of the lower central series of F n . For l ⩾ 1 , we denote by F n Q l the quotient group of F n by a normal subgroup γ 2 ( γ 3 ( F n ) ) γ l + 2 ( γ 2 ( F n ) ) . In this paper, we show that each of the graded quotients of the lower central series of the group F n Q l for any l ⩾ 1 is a free abelian group, and give a basis of it by using a generalized Chen's integration in free groups.