Generalized Stallings' decomposition theorem for pro-$p$ groups

TL;DR

This paper extends Stallings' decomposition theorem to pro-$p$ groups, allowing splittings over infinite pro-$p$ groups, and shows that generalized accessibility is preserved under commensurability.

Contribution

It generalizes the pro-$p$ version of Stallings' theorem to include splittings over infinite pro-$p$ groups, a novel extension not present in the abstract theory.

Findings

Extended Stallings' theorem to infinite pro-$p$ groups

Proved that generalized accessibility is closed under commensurability

No existing abstract analogs for this generalization

Abstract

The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We generalize the pro-$p$ version of it proved by Weigel and the second author to splittings over infinite pro-$p$ groups. This generalization does not have any abstract analogs. We also prove that generalized accessibility of finitely generated pro-$p$ groups is closed for commensurability.