Subgroups of Finitely Presented Centre-by-metabelian Groups

1.1. Baumslag's theorem has another facet. It shows that in the variety 2l of metabelian groups, every finitely generated $I-group G occurs as a subgroup of some finitely presented 2I-group G. The analogous result for the variety of all groups is a celebrated theorem of G. Higman [8] which states that a finitely generated group G is the subgroup of a finitely presented group G if, and only if, G admits a recursive presentation. Recently F. Cannonito [5] rekindled interest in 33-varietal analogues of G. Higman's result for other varieties 33 by proposing that 'analogue' should be interpreted as a characterization of those finitely generated 33-groups that occur as subgroups of finitely presented 33-groups, and by putting forward conjectured analogues for some specific soluble varieties. In particular, he stated a conjecture for the variety (£ of all centre-by-metabelian groups, based on the discovery of J. R. J. Groves [6] that finitely presented C-groups necessarily have the property of being abelian-by-polycyclic, a property that is inherited by subgroups.