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On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the $A$-polynomial

Xiaoyu Xu

TL;DR

This work proves that any isomorphism between the profinite completions of the fundamental groups of cusped finite-volume hyperbolic 3-manifolds is regular and peripheral regular, using a relative profinite (co)homology framework and a profinite mapping degree. By introducing the homology coefficient $\mathbf{coef}(f)$ and establishing the relations $\mathbf{coef}(f)^2=\pm\mathbf{deg}(f)$ and $\mathbf{coef}(f)^3=\pm\mathbf{deg}(f)$, the author deduces that $\mathbf{coef}(f)=\pm1$ and $\mathbf{deg}(f)=\pm1$, hence regularity. Through Dehn filling and boundary analysis, these invariants are propagated to closed manifolds, enabling the main result to apply to knot exteriors and yielding that the enhanced $A$-polynomial $\widetilde{A}_K(M,L)$ is a profinite invariant up to mirror image for prime knots in $S^3$. The paper then extends these ideas to mixed manifolds and derives consequences for profinite detection of JSJ-decomposition and for matching $\mathrm{SL}(n,\mathbb{C})$-representations, thereby connecting peripheral structure, Dehn surgery, and knot invariants in a profinite setting.

Abstract

We prove that any isomorphism between the profinite completions of the fundamental groups of two cusped finite-volume hyperbolic 3-manifolds is regular and peripheral regular. As an application, we show that the $A$-polynomial of prime knots in $S^3$ is a profinite invariant, up to possible mirror image.

On regularity of profinite isomorphisms between cusped hyperbolic 3-manifolds and the $A$-polynomial

TL;DR

This work proves that any isomorphism between the profinite completions of the fundamental groups of cusped finite-volume hyperbolic 3-manifolds is regular and peripheral regular, using a relative profinite (co)homology framework and a profinite mapping degree. By introducing the homology coefficient and establishing the relations and , the author deduces that and , hence regularity. Through Dehn filling and boundary analysis, these invariants are propagated to closed manifolds, enabling the main result to apply to knot exteriors and yielding that the enhanced -polynomial is a profinite invariant up to mirror image for prime knots in . The paper then extends these ideas to mixed manifolds and derives consequences for profinite detection of JSJ-decomposition and for matching -representations, thereby connecting peripheral structure, Dehn surgery, and knot invariants in a profinite setting.

Abstract

We prove that any isomorphism between the profinite completions of the fundamental groups of two cusped finite-volume hyperbolic 3-manifolds is regular and peripheral regular. As an application, we show that the -polynomial of prime knots in is a profinite invariant, up to possible mirror image.

Paper Structure

This paper contains 39 sections, 67 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Regularity and peripheral regularity
  3. The $A$-polynomial
  4. Structure of the paper
  5. Preliminaries on profinite groups
  6. Profinite groups
  7. Residual finiteness
  8. The completion fuctor
  9. Lattice of finite-index subgroups
  10. The profinite integers
  11. $\widehat{\mathbb{Z}}^{\times}$-regularity and peripheral $\widehat{\mathbb{Z}}^{\times}$-regularity
  12. $\widehat{\mathbb{Z}}^{\times}$-regularity
  13. Peripheral $\widehat{\mathbb{Z}}^{\times}$-regularity
  14. The homology coefficient
  15. The group-level explanation
  16. ...and 24 more sections

Key Result

Theorem 1.1

Suppose $M$ and $N$ are cusped finite-volume hyperbolic 3-manifolds and $f:\widehat{\pi_1M}\to \widehat{\pi_1N}$ is an isomorphism. Then $f$ is regular and peripheral regular.

Theorems & Definitions (138)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: RZ10
  • Proposition 2.2
  • Proposition 2.3: RZ10
  • ...and 128 more