Finite presentation of fibre products of metabelian groups

Gilbert Baumslag, Martin R. Bridson, Derek F. Holt, Charles F. Miller III

To appear in J. Pure and Appl. Algebra

We show that if $\G$ is a finitely presented metabelian group, then the ``untwisted" fibre product or pull-back $P$ associated to any short exact sequence $1\to N\to \G\to Q\to 1$ is again finitely presented. In contrast, if $N$ and $Q$ are abelian, then the analogous ``twisted" fibre-product is not finitely presented unless $\G$ is polycyclic. Also a number of examples are constructed, including a non-finitely presented metabelian group $P$ with $H_2(P,\Z)$ finitely generated.