J. Aust. Math. Soc.  74 (2003), 295-312
Group laws implying virtual nilpotence

R. G. Burns
  Department of Mathematics and Statistics
  York University
  Toronto, Ontario
Yuri Medvedev
  Bank of Montreal
  Toronto, Ontario
  Canada M3J 1P3

If $w\equiv 1$ 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 $\mathcal{S}$ of groups including all residually or locally soluble-or-finite groups. In fact the groups of $\mathcal{S}$ 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 $w$ alone. This yields a dichotomy for words. Finally, if the law $w\equiv 1$ 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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