Abstract: Two compact 3-manifolds are profinitely isomorphic if their lattices of finite coverings are isomorphic, or equivalently, their fundamental groups have the same collection of finite quotients. As 3-manifolds are largely determined by their fundamental groups, profinite rigidity in 3-manifolds questions whether the profinite isomorphism type uniquely distinguishes a 3-manifold from all compact (orientable) 3-manifolds.
In fact, profinite rigidity in 3-manifolds is closely related with geometrization. We prove that given a compact orientable 3-manifold M with empty or toral boundary, there are only finitely many compact orientable 3-manifolds profinitely isomorphic to M. In addition, any profinitely isomorphic pair of closed mixed 3-manifolds have homeomorphic Seifert parts in the JSJ-decomposition.
We also prove the profinite rigidity of some mixed 3-manifolds and some cusped hyperbolic 3-manifolds (among compact orientable 3-manifolds), such as the Whitehead link complement. However, profinite rigidity in 3-manifolds does not hold in general. If time permits, we will also introduce some examples of mixed 3-manifolds with non-empty boundary whose fundamental groups are not profinitely rigid.
