Para-G Relations and Hirsch Length in Residually Nilpotent Groups

Abstract

This research explores the interplay between residually nilpotent groups G and H, focusing on their relationship through the lens of para-G conditions and the Hirsch length. We establish criteria for H to be para-G concerning monomorphisms inducing isomorphisms between corresponding lower central quotients of G and H. Specifically, we investigate these conditions in the context of finitely generated residually nilpotent groups. Further, for certain polycyclic groups, we establish connections between para-G relations and the equality of Hirsch lengths. Additionally, we delve into the pro-nilpotent completions of these polycyclic groups, demonstrating their local polycyclic nature.