Groups and spaces with all localizations trivial

The genus of a finitely generated nilpotent group G is defined as the set of isomorphism classes of finitely generated nilpotent groups K such that the p-localizations Kp, Gp are isomorphic for all primes p [19]. This notion turns out to be particularly relevant in the study of non-cancellation phenomena in group theory and homotopy theory. In the above definition, the restriction of finite generation is imposed in order to prevent the genera from becoming too large—in fact, with that restriction, genera are always finite sets. Nevertheless, it is perfectly possible to deal with the so-called extended genus , in which the groups involved, though still nilpotent, are no longer asked to be finitely generated. This generalization has been found to be useful [9, 10, 15]. More serious difficulties arise in this context if one attempts to remove the hypothesis of nilpotency. Given any family of idempotent functors {Ep} in the category of groups, one for each prime p, extending p-localization of nilpotent groups, one could expect to find groups G such that EpG = 1 for all primes p, that is, belonging to the “genus” of the trivial group. In fact, as shown below, there even exist groups G sharing this property for every family {Ep} chosen. We call such groups generically trivial . In Sections 1 and 2 we exhibit their basic properties and point out several sources of examples.