Higher dimensional algebraic fiberings for pro-$p$ groups

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.