Small, $nm$-stable compact $G$-groups

TL;DR

This paper investigates the structure of small, nm-stable compact G-groups, proving they are nilpotent-by-finite or abelian-by-finite under certain conditions, and provides counterexamples to existing conjectures.

Contribution

It establishes new structural results for small, nm-stable compact G-groups and introduces counterexamples to the NM-gap conjecture.

Findings

H is nilpotent-by-finite for small, nm-stable compact G-groups

H is abelian-by-finite if NM(H) ≤ ω

Counterexamples of infinite NM-rank groups challenge the NM-gap conjecture

Abstract

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\NM(H) \leq \omega$, then $H$ is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, $nm$-stable compact $G$-group is abelian-by-finite. We give examples of small, $nm$-stable compact $G$-groups of infinite ordinal $\NM$-rank, providing counter-examples to the $\NM$-gap conjecture.