On some relatively free pro-p groups

TL;DR

This work addresses the FA status of certain relatively free pro-$p$ groups. It develops a framework that assigns first-order axioms to characterize the groups $F/Q$ (the free centre-by-metabelian pro-$p$ group on two generators) and $F/D_{3}$ (the free $\mathfrak{N}_{2}$-by-abelian pro-$p$ group) among profinite groups, relying on a detailed module-theoretic analysis of $M_{1}=D/D_{2}$ and $M_{2}=D_{2}/D_{3}$ over the completed group algebra $R=\mathbb{Z}_{p}[[X]]$ and on centralizer structure. The authors show that these groups are FA in the class of all profinite groups, extending FA results from metabelian pro-$p$ contexts to specific relatively free constructions, and they provide an interpretable bridge to the ring $R$ via the subgroup $D/D_{2}$. The findings contribute a concrete step toward understanding FA properties of free pro-$p$ groups and furnish a methodological blueprint for analyzing FA via Hopfian arguments, centralizers, and completed-group-algebra techniques. The work thus advances both the theory of first-order definability in profinite groups and the structural study of relatively free pro-$p$ groups with constrained nilpotency and abelian quotients.

Abstract

It is shown that the relatively free centre-by-metabelian and (class-2 nilpotent)-by-abelian pro-p groups on 2 generators are each finitely axiomatizable in the class of all profinite groups.

Theorems & Definitions (13)

• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Lemma 1.4
• Theorem 2.1
• Lemma 2.2
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3
• Proposition 3.4