EMBEDDING PROPERTIES OF METABELIAN LIE ALGEBRAS AND METABELIAN DISCRETE GROUPS

Abstract

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.