Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds

TL;DR

This paper demonstrates that certain cusped hyperbolic 3-manifolds are uniquely determined by their profinite completions, using Dehn fillings and characterising slopes to establish profinite rigidity for various examples.

Contribution

It introduces a method to prove profinite rigidity of hyperbolic 3-manifolds by analyzing Dehn fillings and characterising slopes, identifying numerous new profinitely rigid examples.

Findings

Profinite isomorphisms induce isomorphisms between Dehn fillings.

Certain hyperbolic 3-manifolds are uniquely determined by their profinite completions.

Identified multiple classes of profinitely rigid 3-manifolds, including link complements and knots.

Abstract

Any profinite isomorphism between two cusped finite-volume hyperbolic 3-manifolds carries profinite isomorphisms between their Dehn fillings. With this observation, we prove that some cusped finite-volume hyperbolic 3-manifolds are profinitely rigid among all compact, orientable 3-manifolds, through detecting their exceptional Dehn fillings. In addition, we improved a criteria for profinite rigidity of a hyperbolic knot complement or a hyperbolic-type satellite knot complement among compact, orientable 3-manifolds, through examining its characterising slopes. We obtain the following profinitely rigid examples: the complement of the Whitehead link, Whitehead sister link, $\frac{3}{10}$ two-bridge link; specific surgeries on one component of these links; the complement of (full) twist knots $\mathcal{K}_n$, Eudave-Mu\~noz knots $K(3,1,n,0)$, Pretzel knots $P(-3,3,2n+1)$, $5_2$ knot; the Berge manifold, and many more.