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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 by1    01    1,x    00    1,n  !00    1,i  +  x00    1for    i  =  1  ,  ⋯  ,  n    is of type    F   P   n  +  1but not of type    F   P   n  +  2, thus providing evidence in favour of the Bieri–Groves    F   P  m    ‐conjecture. These examples are amongst the simplest known examples of torsion‐free metabelian groups of their kind.

Homological finiteness conditions for a class of metabelian groups