J. Aust. Math. Soc.
74 (2003), 295-312
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Group laws implying virtual nilpotence
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Yuri Medvedev
Bank of Montreal
Toronto, Ontario
Canada M3J 1P3
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Abstract
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If  is a group law implying virtual nilpotence in
every finitely generated metabelian group
satisfying it, then it implies virtual nilpotence
for the finitely generated groups of a large
class  of groups including all residually or locally
soluble-or-finite groups. In fact the groups
of satisfying such a law are all
nilpotent-by-finite exponent where the nilpotency
class and exponent in question are both bounded
above in terms of the length
of  alone. This yields a dichotomy for words.
Finally, if the
law satisfies a certain additional
condition---obtaining in particular for any
monoidal or Engel law---then the conclusion
extends to the much larger class consisting of
all `locally graded' groups.
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