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

Xiaoyu Xu

TL;DR

This work advances profinite rigidity for cusped finite-volume hyperbolic 3-manifolds by proving that any profinite isomorphism induces a correspondence between Dehn fillings via peripheral $\widehat{\mathbb{Z}}^{\times}$-regularity. The authors develop a robust framework to detect exceptional Dehn fillings from finite quotients, linking prime decomposition, toroidality, and Klein bottles to profinite data. They establish explicit profinite rigidity for a variety of manifolds, including the Whitehead link complement, Whitehead sister link, the $\tfrac{3}{10}$-two-bridge link, several twist and Eudave-Muñoz knots, the Berge manifold, and related surgeries, often via characterising slopes. The results provide both new rigid examples and refined criteria for knot-complement rigidity, suggesting that profinite information may strongly constrain hyperbolic 3-manifolds and supporting the broader conjecture that profinite data uniquely determines a cusped hyperbolic 3-manifold within the orientable category.

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-Muñoz knots $K(3,1,n,0)$, Pretzel knots $P(-3,3,2n+1)$, $5_2$ knot; the Berge manifold, and many more.

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

TL;DR

This work advances profinite rigidity for cusped finite-volume hyperbolic 3-manifolds by proving that any profinite isomorphism induces a correspondence between Dehn fillings via peripheral -regularity. The authors develop a robust framework to detect exceptional Dehn fillings from finite quotients, linking prime decomposition, toroidality, and Klein bottles to profinite data. They establish explicit profinite rigidity for a variety of manifolds, including the Whitehead link complement, Whitehead sister link, the -two-bridge link, several twist and Eudave-Muñoz knots, the Berge manifold, and related surgeries, often via characterising slopes. The results provide both new rigid examples and refined criteria for knot-complement rigidity, suggesting that profinite information may strongly constrain hyperbolic 3-manifolds and supporting the broader conjecture that profinite data uniquely determines a cusped hyperbolic 3-manifold within the orientable category.

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, two-bridge link; specific surgeries on one component of these links; the complement of (full) twist knots , Eudave-Muñoz knots , Pretzel knots , knot; the Berge manifold, and many more.

Paper Structure

This paper contains 25 sections, 51 theorems, 15 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Profinite groups
  4. Residual finiteness
  5. Profinite detection of hyperbolicity
  6. Peripheral structure
  7. Peripheral $\widehat{\mathbb{Z}}^{\times}$-regularity and correspondence of Dehn fillings
  8. Peripheral $\widehat{\mathbb{Z}}^{\times}$-regularity
  9. Dehn filling
  10. Knot and link complements
  11. Detection of knot and link complements
  12. Identifying Dehn surgery coefficients
  13. Criteria for profinite rigidity of a knot complement
  14. Profinite detection of exceptional Dehn fillings
  15. Detecting prime decomposition and asphericity
  16. ...and 10 more sections

Key Result

Theorem A

Suppose $M$ and $N$ are orientable cusped finite-volume hyperbolic 3-manifolds, and $\widehat{\pi_1M}\cong \widehat{\pi_1N}$. Then there exists a homeomorphism $\Psi: \partial M\to \partial N$ such that for any boundary slopes $(c_i)$ on $\partial M$ (allowing empty slopes), there is an isomorphism

Figures (11)

  • Figure 1: Link diagrams for $\mathcal{W}$, $\mathcal{WS}$, $\mathcal{L}$
  • Figure 2: Knot diagrams for $\mathcal{K}_n$
  • Figure 3: Knot diagram for $\mathcal{J}_n$
  • Figure 4: Knot diagram for $K(3,1,n,0)$
  • Figure 5: Link diagrams for $L_A$ and $L_B$
  • ...and 6 more figures

Theorems & Definitions (106)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem C
  • Definition 1.4
  • ...and 96 more