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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.