Profinite almost rigidity in 3-manifolds

TL;DR

This paper demonstrates that the profinite completion of the fundamental group of certain 3-manifolds uniquely determines their homeomorphism type, highlighting a form of almost rigidity in the profinite setting.

Contribution

It establishes profinite almost rigidity for compact orientable 3-manifolds and clarifies the role of peripheral structures in determining Seifert parts.

Findings

Profinite completion determines the manifold's homeomorphism type up to finitely many options.

Peripheral structure is crucial for uniquely identifying Seifert parts.

Without peripheral data, the profinite completion may not distinguish Seifert components.

Abstract

We prove that any compact, orientable 3-manifold with empty or toral boundary is profinitely almost rigid among all compact, orientable 3-manifolds. In other words, the profinite completion of its fundamental group determines its homeomorphism type to finitely many possibilities. Moreover, the profinite completion of the fundamental group of a mixed 3-manifold, together with the peripheral structure, uniquely determines the homeomorphism type of its Seifert part, i.e. the maximal graph manifold components in the JSJ-decomposition. On the other hand, without assigning the peripheral structure, the profinite completion of a mixed 3-manifold group may not uniquely determine the fundamental group of its Seifert part. The proof is based on JSJ-decomposition.