Profinite almost rigidity in 3-manifolds

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.