Free Groups of Outer Commutator Varieties of Groups

If F is a free group, 1 < i j 2i and i k i + j + 1 thenF/_j(F), _i(F), _k(F)] is residually nilpotent and torsion-free.This result is extended to 1 < i j 2i and i k 2i + 2j.It is proved that the analogous Lie rings, L/L^j, Lⁱ, L^k] whereL is a free Lie ring, are torsion-free. Candidates are foundfor torsion in L/L^j, Lⁱ, L^k] whenever k is the least of {i,j, k}, and the existence of torsion in L/L^j, Lⁱ, L^k] is provedwhen i, j, k 5 and k is the least of {i, j, k}.