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Classification of Hyperbolic Dehn fillings II: Quadratic case

BoGwang Jeon

TL;DR

The paper delivers a complete, effective classification for hyperbolic Dehn fillings in the quadratic-field cusp setting, introducing a rigorous automorphism framework for the holonomy variety and leveraging the pseudo complex volume to bound equivalence classes of fillings. It develops a finite-group, linear-algebraic approach (Types I–III, primary matrices) to classify automorphisms of the analytic holonomy set, with special attention to SGI cusps and cubic/quartic minimal polynomials. The results yield explicit bounds on the number of fillings sharing pseudo complex volume, dependent on cusp fields, and reveal underlying structures through concrete examples (notably v2788) and the Zilber–Pink-type constraints in the analytic setting. The work lays groundwork for CHDFIII, where optimality and broader instances are illustrated, combining number-theoretic ideas with linear-algebraic analysis to advance understanding of the complex volume landscape across Dehn fillings.

Abstract

This paper is subsequent to [5]. In this paper, we extend the classification of hyperbolic Dehn fillings with sufficiently large coefficients by addressing the remaining case not covered in [5]. Specifically, by considering the case in which the two cusp shapes lie in the same quadratic field, we obtain the complete classification under a mild assumption satisfied by most manifolds. The content of this paper is not limited to the classification of hyperbolic Dehn fillings. Along the way, we also classify key types of automorphisms of the holonomy variety of a two-cusped hyperbolic $3$-manifold and uncover an intriguing hidden structure in the complex volume of certain manifolds. Concrete examples illustrating these phenomena were discovered by S. Oh, and we elaborate on them in this paper. All the results presented here appear to be effective. In the third paper of this series, joint with S. Oh, we will provide examples confirming the optimality of our results.

Classification of Hyperbolic Dehn fillings II: Quadratic case

TL;DR

The paper delivers a complete, effective classification for hyperbolic Dehn fillings in the quadratic-field cusp setting, introducing a rigorous automorphism framework for the holonomy variety and leveraging the pseudo complex volume to bound equivalence classes of fillings. It develops a finite-group, linear-algebraic approach (Types I–III, primary matrices) to classify automorphisms of the analytic holonomy set, with special attention to SGI cusps and cubic/quartic minimal polynomials. The results yield explicit bounds on the number of fillings sharing pseudo complex volume, dependent on cusp fields, and reveal underlying structures through concrete examples (notably v2788) and the Zilber–Pink-type constraints in the analytic setting. The work lays groundwork for CHDFIII, where optimality and broader instances are illustrated, combining number-theoretic ideas with linear-algebraic analysis to advance understanding of the complex volume landscape across Dehn fillings.

Abstract

This paper is subsequent to [5]. In this paper, we extend the classification of hyperbolic Dehn fillings with sufficiently large coefficients by addressing the remaining case not covered in [5]. Specifically, by considering the case in which the two cusp shapes lie in the same quadratic field, we obtain the complete classification under a mild assumption satisfied by most manifolds. The content of this paper is not limited to the classification of hyperbolic Dehn fillings. Along the way, we also classify key types of automorphisms of the holonomy variety of a two-cusped hyperbolic -manifold and uncover an intriguing hidden structure in the complex volume of certain manifolds. Concrete examples illustrating these phenomena were discovered by S. Oh, and we elaborate on them in this paper. All the results presented here appear to be effective. In the third paper of this series, joint with S. Oh, we will provide examples confirming the optimality of our results.

Paper Structure

This paper contains 52 sections, 69 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Main result I
  3. Refinement of Theorem \ref{['23070303']}
  4. Main result II
  5. Refinement of Theorem \ref{['22050203']}
  6. How to classify $H$?
  7. Remarks on the approach
  8. Outline of the paper
  9. Acknowledgement
  10. Preliminaries
  11. Essence from the precedent
  12. Holonomy variety: algebraic & analytic
  13. Dehn filling points
  14. Auxiliary theorems I
  15. Auxiliary theorems II
  16. ...and 37 more sections

Key Result

Theorem 1.1

CHDF Let $\mathcal{M}$ be a two-cusped hyperbolic $3$-manifold.Here and throughout the paper, all hyperbolic $3$-manifolds are assumed to be orientable and of finite volume. There exists a finite set $\mathcal{S}(\subset \mathrm{GL}_2(\mathbb{Q})\times \mathrm{GL}_2(\mathbb{Q}))$ such that, for any $(p'_1/q'_1, p'_2/q'_2)$ is equal to either for some $(\sigma, \rho)\in \mathcal{S}$. Further, if

Theorems & Definitions (126)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 1.7: S. Oh
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 116 more