J. Aust. Math. Soc.
74 (2003), 295312

Group laws implying virtual nilpotence



Yuri Medvedev
Bank of Montreal
Toronto, Ontario
Canada M3J 1P3



Abstract

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
solubleorfinite groups. In fact the groups
of satisfying such a law are all
nilpotentbyfinite 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
conditionobtaining in particular for any
monoidal or Engel lawthen the conclusion
extends to the much larger class consisting of
all `locally graded' groups.

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