On pro-$p$ analogues of limit groups via extensions of centralizers

TL;DR

This paper introduces a new class of pro-$p$ groups called $\\mathcal{L}$, inspired by limit groups, and explores their algebraic and cohomological properties, revealing structural constraints and classifications.

Contribution

It defines the class $\\mathcal{L}$ of pro-$p$ groups via extensions of centralizers and establishes key properties including cohomological finiteness, structural decompositions, and classification of 2-generated groups.

Findings

Groups in $\\mathcal{L}$ have finite cohomological dimension.

They are free-by-(torsion-free poly-procyclic).

Every 2-generated group in $\\mathcal{L}$ is either free pro-$p$ or abelian.

Abstract

We begin a study of a pro-$p$ analogue of limit groups via extensions of centralizers and call $\mathcal{L}$ this new class of pro-$p$ groups. We show that the pro-$p$ groups of $\mathcal{L}$ have finite cohomological dimension, type $FP_{\infty}$ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion-free poly -procyclic) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-$p$ group in the class $\mathcal{L}$ is either free pro-$p$ or abelian.