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Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let γn(F) be the nth term of the lower central series of F. It is proved that F/[γj(F), γi(F), γk(F)] and F/[γj(F), γi(F), γk(F), γl(F)] are torsion free and residually nilpotent for certain values of i, j, k and i, j, k, l, respectively. In the process of proving this, it is proved that the analogous Lie rings are torsion free.