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Profinite rigidity and geometric convergence

Yu Huang

TL;DR

This work shows that profinite rigidity for finite-volume hyperbolic 3-manifolds is preserved under geometric limits, by leveraging Dehn-filling compatibility of profinite data and a bubbling construction. The authors introduce bubble-drilling on fibered 3-manifolds, establish a practical hyperbolicity criterion via flow-acoannularity, and prove that if all fibered hyperbolic manifolds with a fixed fiber are profinitely rigid, then the resulting bubble-drilled manifolds are as well. Consequently, the set of profinitely rigid cusped manifolds is shown to be closed in the geometric topology, and several explicit examples, including Whitehead link, Borromean ring, and a 5-chain link complement, are proven profinitely rigid in the ambient class $\mathfrak{M}$. The results reduce cusped cases to closed ones under certain conditions and provide a robust framework for generating new profinitely rigid hyperbolic 3-manifolds with intricate cusp structures.

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.

Profinite rigidity and geometric convergence

TL;DR

This work shows that profinite rigidity for finite-volume hyperbolic 3-manifolds is preserved under geometric limits, by leveraging Dehn-filling compatibility of profinite data and a bubbling construction. The authors introduce bubble-drilling on fibered 3-manifolds, establish a practical hyperbolicity criterion via flow-acoannularity, and prove that if all fibered hyperbolic manifolds with a fixed fiber are profinitely rigid, then the resulting bubble-drilled manifolds are as well. Consequently, the set of profinitely rigid cusped manifolds is shown to be closed in the geometric topology, and several explicit examples, including Whitehead link, Borromean ring, and a 5-chain link complement, are proven profinitely rigid in the ambient class . The results reduce cusped cases to closed ones under certain conditions and provide a robust framework for generating new profinitely rigid hyperbolic 3-manifolds with intricate cusp structures.

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.
Paper Structure (13 sections, 17 theorems, 1 equation, 7 figures)

This paper contains 13 sections, 17 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Profinite completion
  4. Dehn fillings
  5. Profinite rigidity and geometric convergence
  6. Geometric topology and geometric convergence
  7. Profinite rigidity of a geometric limit
  8. Reduce cusped cases to closed cases
  9. The bubbling trick
  10. The hyperbolicity criterion
  11. New examples for profinite rigidity
  12. (1) The origin manifold $M$ is the figure-eight knot complement.
  13. (2) The origin manifold $M$ is a three-punctured disk bundle over $S^{1}$

Key Result

Theorem 1.1

If $M$ and $N$ are two profinitely isomorphic finite-volume hyperbolic 3-manifolds with $k>0$ cusps, then there exists a homeomorphism $h: \partial M\to \partial N$ such that $\widehat{\pi_{1}M_{\gamma}}\cong \widehat{\pi_{1}N_{h(\gamma)}}$ for any Dehn filling parameters $\gamma=(\gamma_{1},\ldots,

Figures (7)

  • Figure 1: Three links noted in Theorem \ref{['cor:example']}. They are the Whitehead link, the Borreanmean ring, the specific 5-chain link, from left to right respectively.
  • Figure 2: Annuli foliation of a solid torus after removing highest and lowest two small arches.
  • Figure 3: A fiber of the fibration of figure-right knot
  • Figure 4: A bubbled-drilled manifold homeomorphic to Whitehead link complement
  • Figure 5: A bubbled-drilled manifold homeomorphic to Borromean ring complement
  • ...and 2 more figures

Theorems & Definitions (41)

  • Theorem 1.1: Xu24b
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Proposition 2.2: WZ17aWZ17bWZ19a
  • Proposition 2.3
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 31 more