Higher dimensional algebraic fiberings for pro-$p$ groups

TL;DR

This paper investigates conditions under which pro-$p$ groups exhibit higher dimensional algebraic fibering, leading to new insights on their incoherence and structural properties, especially in relation to free pro-$p$ groups and their automorphisms.

Contribution

It establishes new conditions for higher dimensional algebraic fibering of pro-$p$ groups and demonstrates incoherence results, extending classical group theory results to the pro-$p$ setting.

Findings

Pro-$p$ groups with specific extensions are shown to be incoherent.

Automorphism groups of free pro-$p$ groups can be incoherent.

A pro-$p$ analogue of Bieri-Strebel's result does not hold in certain cases.

Abstract

We prove some conditions for higher dimensional algebraic fibering of pro-$p$ group extensions and we establish corollaries about incoherence of pro-$p$ groups. In particular, if $G = K \rtimes \Gamma$ is a pro-$p$ group, $\Gamma$ a finitely generated free pro-$p$ group with $d(\Gamma) \geq 2$, $K$ a finitely presented pro-$p$ group with $N$ a normal pro-$p$ subgroup of $K$ such that $K/ N \simeq \mathbb{Z}_p$ and $N$ not finitely generated as a pro-$p$ group, then $G$ is incoherent (in the category of pro-$p$ groups). Furthermore we show that if $K$ is a free pro-$p$ group with $d(K) = 2$ then either $Aut_0(K)$ is incoherent (in the category of pro-$p$ groups) or there is a finitely presented pro-$p$ group, without non-procyclic free pro-$p$ subgroups, that has a metabelian pro-$p$ quotient that is not finitely presented i.e. a pro-$p$ version of a result of Bieri-Strebel does not hold.