Profinite completions, cohomology and JSJ decompositions of compact 3-manifolds

TL;DR

This paper extends the understanding of how JSJ decompositions of compact 3-manifolds with boundary behave under profinite completions, introducing a cohomological approach to analyze group actions on profinite trees.

Contribution

It generalizes previous results to manifolds with boundary and offers a new cohomological method for studying group actions in this context.

Findings

Profinite completions reflect JSJ decompositions of compact 3-manifolds with boundary.

Relative cohomology provides a natural framework for analyzing group actions.

New insights into the structure of 3-manifold groups via profinite methods.

Abstract

In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative cohomology to analyse the actions of an-annular atoroidal groups on profinite trees.