The nilpotent genus of finitely generated residually nilpotent groups

Abstract

Abstract Let 𝐺 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 𝐻 is para-𝐺. We then prove that, for certain polycyclic groups, if 𝐻 is para-𝐺, then 𝐺 and 𝐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.