Universal epimorphic equivalences for group localizations

Recent work by Bousfield shows the existence, for any map φ, of a universal space that is killed by homotopical φ-localization. Nullification with respect to this so-called universal φ-acyclic space is related to φ-localization in the same way as Quillenʹs plus construction is related to homological localization. Here we construct a universal f-acyclic group for any group homomorphism f. Moreover, we prove that there is a universal epimorphism that is inverted by f-localization. Although the kernel of the -localization homomorphism coincides with that of the f-localization homomorphism, we show that localization with respect to has in general nicer properties than f-localization itself.