Absolute Closedness of Torsion-Free Abelian Groups in the Class of Metabelian Groups

The dominion of a subgroup H of a group G in a class M is the set of all elements a a G whose images are equal for all pairs of homomorphisms from G to each group in M that coincide on H. A group H is absolutely closed in a class M if, for any group G in M, every inclusion H a parts per thousand currency sign G implies that the dominion of H in G (in M) coincides with H. We deal with dominions in torsion-free Abelian subgroups of metabelian groups. It is proved that every nontrivial torsion-free Abelian group is not absolutely closed in the class of metabelian groups. It is stated that if a torsion-free subgroup H of a metabelian group G and the commutator subgroup G' have trivial intersection, then the dominion of H in G (in the class of metabelian groups) coincides with H.