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FREE GROUPS OF OUTER COMMUTATOR VARIETIES OF GROUPS

Published online by Cambridge University Press:  30 October 2001

DANIEL P. GROVES
Affiliation:
Merton College, Oxford OX1 4JD; [email protected]
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Abstract

If F is a free group, 1 < i [les ] j [les ] 2i and i [les ] k [les ] i + j + 1 then F/[γj(F), γi(F), γk(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/[Lj, Li, Lk] where L is a free Lie ring, are torsion-free. Candidates are found for torsion in L/[Lj, Li, Lk] whenever k is the least of {i, j, k}, and the existence of torsion in L/[Lj, Li, Lk] is proved when i, j, k [les ] 5 and k is the least of {i, j, k}.

Type
Research Article
Copyright
The London Mathematical Society 2001

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