Improving Finiteness Properties for Metabelian Groups

We show that any finitely generated metabelian group can be embedded in a metabelian group of type F3. The proof builds upon work of G. Baumslag [4], who independently with V. R. Remeslennikov [10] 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 [7], 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 ZQ-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 [6] to imply the F3 embedding theorem. AMS Subject classification: Primary 20F16, Secondary 20J06.