Pro-$p$ subgroups of profinite completions of 3-manifold groups

TL;DR

This paper characterizes finitely generated pro-$p$ subgroups within the profinite completions of 3-manifold groups and extends Bass-Serre theory to pro-$p$ groups, advancing understanding of their algebraic structure.

Contribution

It provides a complete description of these subgroups and introduces a pro-$p$ analogue of Bass-Serre theory for finitely generated pro-$p$ groups.

Findings

Complete classification of finitely generated pro-$p$ subgroups

Pro-$p$ analogue of Bass-Serre theory established

Enhanced understanding of 3-manifold group completions

Abstract

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.