The nilpotent genus of finitely generated residually nilpotent groups
Niamh O'Sullivan

TL;DR
This paper investigates the relationships between finitely generated residually nilpotent groups, focusing on conditions for para-$G$ groups and their properties, including Hirsch length and pro-nilpotent completions.
Contribution
It establishes sufficient conditions for a group to be para-$G$ and explores properties of polycyclic groups related to their nilpotent genus and pro-nilpotent completions.
Findings
Conditions for a group to be para-$G$ are identified.
Para-$G$ groups share the same Hirsch length in certain cases.
Pro-nilpotent completions of specific polycyclic groups are locally polycyclic.
Abstract
If and are finitely generated residually nilpotent groups, then and are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that is para- if there exists a monomorphism of into which induces isomorphisms between the corresponding quotients of their lower central series. We first consider residually nilpotent groups and find sufficient conditions on the monomorphism so that is para- We then prove that for certain polycyclic groups, if is para-, then and have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.
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Taxonomy
TopicsFinite Group Theory Research · Geometric and Algebraic Topology
