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

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.

The profinite completions of knot groups determine the Alexander polynomials

“…By Funar ([Fun13]) and Hempel ([Hem14]), there are pairs of torus bundles and those of Seifert 3-manifolds whose fundamental groups are not isomorphic but whose π's are isomorphic, while the existence of such a pair of hyperbolic 3-manifolds is still unknown. Other recent progresses are due to Wilton-Zalesskii and Wilkes ([WZ10], [WZ17b], [Wil17a], [Wil18b], [Wil17b], [Wil18c], [Wil18a]).…”