Profinite rigidity and geometric convergence

In this paper, we prove that profinitely rigid finite-volume hyperbolic manifolds form a closed set under geometric topology. This observation implies the profinite rigidity of a large family of cusped hyperbolic manifolds via bubble-drilling construction. The core of the proof is a strong criterion that is used to verify when bubble-drilled manifolds are hyperbolic. This family includes many link complements, such as the Whitehead link complement and the Borromean ring complement.