On the lower central quotients of the free groups of finite rank of a certain variety. (Q716475)

In this paper the author examines the graded quotient of the group \(F_n^{\mathfrak D_\ell}=F_n/N\) where \(F_n\) is a free group of rank \(n\) and \(N=\gamma_2(\gamma_3(F_n))\gamma_{\ell+2}(\gamma_2(F_n))\). Here \(\gamma_k(G)\) denotes the \(k\)-th term of the lower central series of \(G\) and the graded quotients are the Abelian groups \(\mathcal L_k(G)=\gamma_k(G)/\gamma_{k+1}(G)\) for \(k\geq 1\). The main result of this paper is that these quotients for \(F_n^{\mathfrak D_\ell}\) are free Abelian groups and gives explicit bases for them. The treatment is combinatorial and for the proofs the author goes back to P. Hall's basic commutators and Chen's integration in free groups [\textit{K.-T. Chen}, Ann. Math. (2) 54, 147-162 (1951; Zbl 0045.30102)].