Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation

TL;DR

This paper studies how finitely generated pro-$p$ groups acting on pro-$p$ trees can be decomposed into fundamental pro-$p$ groups of finite graphs, providing bounds and criteria related to their structure and cohomology.

Contribution

It proves that such groups split over edge stabilizers and are fundamental groups of finite graphs of pro-$p$ groups, with bounds when edge stabilizers are procyclic, and offers a cohomological criterion for accessibility.

Findings

Groups split as free amalgamated products or HNN-extensions.

Finite graphs of pro-$p$ groups describe the structure of these groups.

Bound on the graph size when edge stabilizers are procyclic.

Abstract

We prove that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(\cal G, \Gamma)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $\Gamma$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $G$ to be accessible in terms of the first cohomology $H^1(G, \mathbb{F}_p[[G]])$.