Profinite rigidity and geometric convergence

TL;DR

This paper demonstrates that profinitely rigid finite-volume hyperbolic manifolds are closed under geometric limits and establishes profinite rigidity for many cusped hyperbolic manifolds using a new criterion for hyperbolicity after bubble drilling.

Contribution

It introduces a strong criterion for hyperbolicity of bubble-drilled manifolds and proves the closure of profinitely rigid hyperbolic manifolds under geometric topology.

Findings

Profinite rigidity is preserved under geometric limits for certain hyperbolic manifolds.

Many link complements, including Whitehead and Borromean rings, are profinitely rigid.

A new criterion helps verify hyperbolicity after bubble drilling.

Abstract

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.