W. A. Bogley and J. Harlander, Improving finiteness properties for
metabelian groups, submitted for publication.
Abstract
We show that any finitely generated metabelian group can be
embedded in a metabelian group of type F$_3$. The proof builds upon work of
G.~Baumslag \cite{Ba}, who independently with V.~R.~Remeslennikov \cite{Re2}
proved that any finitely generated metabelian group can be embedded in a
finitely presented one. We also rely essentially on the Sigma theory of
R.~Bieri and R.~Strebel \cite{BiSt}, who introduced this geometric theory to
detect finite presentability of metabelian groups. Within the context of Sigma
theory, we also prove that if $n$ is a positive integer and $Q$ is a finitely
generated abelian group, then any finitely generated $\Z Q$-module can be
embedded in a module that is $n$-tame. For $n=3$, this tameness result
combines with a recent theorem of Bieri and J.~Harlander \cite{BiHa} to imply
the F$_3$ embedding theorem.
AMS Subject classification: Primary 20F16, Secondary 20J06.
Keywords: metabelian group, homological finiteness properties, Sigma
theory, tame module