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Journal of the London Mathematical Society 2006 73(2):475-492; doi:10.1112/S0024610705022581
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© The London Mathematical Society

Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

J. R. J. Groves and D. H. Kochloukova

Department of Mathematics and Statistics, University of Melbourne Parkville, Australia
IMECC UNICAMP, Cx. P. 6065, 13083-970 Campinas, SP, Brazil

Received 26 February 2004. Revision received 8 March 2004.

We show that for every natural number m a finitely generated metabelian group G embeds in a quotient of a metabelian group of type FPm. Furthermore, if m ≤ 4, the group G can be embedded in a metabelian group of type FPm. For L a finitely generated metabelian Lie algebra over a field K and a natural number m we show that, provided the characteristic p of K is 0 or p > m, then L can be embedded in a metabelian Lie algebra of type FPm. This result is the best possible as for 0 < p ≤ m every metabelian Lie algebra over K of type FPm is finite dimensional as a vector space.


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