Commutator subgroups of free nilpotent groups

For fixed m, n >= 2, we examine the structure of the nth lower central subgroup gamma(n)(F) of the free group F of rank m with respect to a certain finite chain F = F-( 0) > F-(1) >center dot center dot center dot > F( l- 1) > F-( l) = {1} of free groups in which F-( k) is of finite rank m( k) and is contained in the kth derived subgroup delta(k)(F) of F. The derived subgroups delta(k)( F/gamma(n)( F)) of the free nilpotent group F/gamma(n)( F) are isomorphic to the quotients F-(k)/( F-(k) boolean AND gamma(n)( F)) and admit presentations of the form < xk, 1,..., xk, m(k) : gamma(n)( F-(k))>, where gamma(n)( F-(k)), contained in gamma(n)( F), is a certain partial lower central subgroup of F-(k). We give a complete description of gamma(n)( F) as a staggered product Pi 1 <= k <= l-1(gamma < n >( F-(k)) *gamma[ n]( F-(k))) F(k+ 1), where gamma < n >( F-(k)) is a free factor of the derived subgroup [ F-(k), F-(k)] of F-(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and gamma([n])( F-(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.