FREE GROUPS OF OUTER COMMUTATOR VARIETIES OF GROUPS

Abstract

If F is a free group, 1 < i [les ] j [les ] 2i and i [les ] k [les ] i + j + 1 then F/[γ_j(F), γ_i(F), γ_k(F)] is residually nilpotent and torsion-free. This result is extended to 1 < i [les ] j [les ] 2i and i [les ] k [les ] 2i + 2j. It is proved that the analogous Lie rings, L/[L^j, Lⁱ, L^k] where L is a free Lie ring, are torsion-free. Candidates are found for 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 proved when i, j, k [les ] 5 and k is the least of {i, j, k}.