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 $\textit{FP}_m$. Furthermore, if $m \leq 4$, the group G can be embedded in a metabelian group of type $\textit{FP}_m$. 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 $\textit{FP}_m$. This result is the best possible as for $0 < p\leq m$ every metabelian Lie algebra over K of type $\textit{FP}_m$ is finite dimensional as a vector space.