In this talk, I will discuss the almost profinite rigidity among the fundamental group of compact orientable 3-manifolds with empty or toral boundary.
I will also introduce some examples of cusped hyperbolic and mixed 3-manifolds whose fundamental groups are profinitely rigid among all 3-manifold groups. The key ingredient is a property called "peripheral regularity" for profinite isomorphisms between cusped hyperbolic 3-manifolds.
