Homological finiteness conditions for a class of metabelian groups

We generalize a theorem of Groves and Kochloukova concerning cohomological finiteness conditions for metabelian groups in order to encompass a classical example of Baumslag and Stammbach. More precisely we shall show that for any natural number n and indeterminate x, the group of 2×2 matrices generated by 1011,x001,n!001,i+x001for i=1,⋯,n is of type FPn+1 but not of type FPn+2, thus providing evidence in favour of the Bieri–Groves FPm‐conjecture. These examples are amongst the simplest known examples of torsion‐free metabelian groups of their kind.