A pro-$p$ version of Sela's accessibility and Poincar\'e duality pro-$p$ groups

TL;DR

This paper extends Sela's accessibility theorem to pro-$p$ groups, demonstrating that finitely generated pro-$p$ groups are $k$-acylindrically accessible and establishing a unique JSJ-decomposition for Poincaré duality pro-$p$ groups.

Contribution

It introduces a pro-$p$ version of Sela's theorem and applies it to prove the uniqueness of JSJ-decomposition for Poincaré duality pro-$p$ groups.

Findings

Pro-$p$ groups are $k$-acylindrically accessible.

Unique $k$-acylindrical JSJ-decomposition exists for $ ext{PD}^n$ pro-$p$ groups.

Extension of Sela's theorem to the pro-$p$ setting.

Abstract

We prove a pro-$p$ version of Sela's theorem stating that a finitely generated group is $k$-acylindrically accessible. This result is then used to prove that $\mathrm{PD}^n$ pro-$p$ groups admit a unique $k$-acylindrical JSJ-decomposition.