Virtually nilpotent groups with (almost) all localizations trivial

A group G is generically trivial if and only if, for all prime numbers p, t he localization of G with respect to p is trivial. Taking off from a theore m of CASACUBERTA and CASTELLET, we prove that a virtually nilpotent group E is generically trivial if and only if E is perfect. Inspired by this result, we introduce the concept of almost generically tri vial groups. Those are groups G such that, for only finitely many primes p, the localization of G with respect to p is not trivial. We prove that a vi rtually nilpotent group E with finitely generated abelianization is almost generically trivial if and only if the abelianization of E is finite.