Finite Presentation of Fibre Products of Metabelian Groups

We show that if Γ is a finitely presented metabelian group, then the “untwisted” fibre product or pull-back P associated to any short exact sequence 1 → N → Γ → Q → 1 is again finitely presented. In contrast, if N and Q are abelian, then the analogous “twisted” fibre-product is not finitely presented unless Γ is polycyclic. Also a number of examples are constructed, including a non-finitely presented metabelian group P with H2(P,Z) finitely generated. Associated to each pair of short exact sequences of groups 1→ Ni → Γi pi → Q → 1, i = 1, 2, one has the fibre product P = {(γ1, γ2) ∈ Γ1×Γ2 | p1(γ1) = p2(γ2)}. In this article we shall be concerned entirely with the case Γ1 = Γ2 = Γ, N1 = N2 = N , and for the most part we shall focus on the case where p1 = p2, where we shall call the fibre product untwisted. We are interested in the question of when such fibre products are finitely presented. There have recently been several significant results in this direction. Firstly, if Γ is free and both Q and N are infinite, then P is never finitely presented [2, 3]. Likewise if Γ is a surface group [6]. On the other hand, if p1 = p2 and one knows that N is finitely generated, Γ is finitely presented and Q is of type F3, then P is always finitely presented — this is the 1-2-3 Theorem of [7]. Intrigued by this contrast in behaviour, we shall look at a class of groups Γ that are far from free and which do not fall within the scope of the 1-2-3 Theorem, namely short exact sequences of metabelian groups. In this context one also finds a contrast in the behaviour of fibre products, even within examples that, superficially, appear very similar Date: February 11, 2002. 1991 Mathematics Subject Classification. Primary 20.