On a problem of Bousfield for metabelian groups

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for R=Q or R=Z/n and a finitely presented metabelian group G, the natural map from G to its R-completion induces an epimorphism of homology groups H2(−,R). This answers a problem of A.K. Bousfield for the class of metabelian groups.