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Hyperbolic Dehn filling, volume, and transcendentality

BoGwang Jeon, Sunul Oh

Abstract

Let $M$ be a 1-cusped hyperbolic 3-manifold. In this paper, we study the behavior of $N_M(v)$, the number of Dehn fillings of $M$ with a given volume $v(\in \mathbb{R})$. We conduct extensive computational experiments to estimate $N_M$ and propose a theoretical framework to explain its behavior. Further, we prove that the growth of $N_M$ is slower than any power of its filling coefficient.

Hyperbolic Dehn filling, volume, and transcendentality

Abstract

Let be a 1-cusped hyperbolic 3-manifold. In this paper, we study the behavior of , the number of Dehn fillings of with a given volume . We conduct extensive computational experiments to estimate and propose a theoretical framework to explain its behavior. Further, we prove that the growth of is slower than any power of its filling coefficient.
Paper Structure (21 sections, 35 theorems, 126 equations, 3 figures, 1 table)

This paper contains 21 sections, 35 theorems, 126 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Experimental results
  3. Theoretical predictions & confirmations
  4. Pila–Wilkie
  5. Application
  6. Outline of the paper
  7. Preliminaries
  8. Holonomy variety: algebraic & analytic
  9. Neumann-Zagier volume formula
  10. Complex length and volume
  11. Proofs of main I and II
  12. Proof of Theorem \ref{['25031903']}
  13. Proof of Theorem \ref{['25041802']}
  14. Experiments
  15. O-minimality and Transcendentality
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M$ be a $1$-cusped hyperbolic 3-manifold and $M(p, q)$ be its $(p, q)$-Dehn filling. Then $M(p, q)$ is hyperbolic for sufficiently large $|p|+|q|$ and as $|p|+|q|$ goes to $\infty$.

Figures (3)

  • Figure 1: The manifold m208 which is obtained from the exterior of the link $6_1^3$ by filling two cusps
  • Figure 2: The manifold m135 which is obtained from the exterior of the link $8_9^3$ by filling two cusps
  • Figure 3: The manifold m009 which is obtained from the Whitehead link complement by filling one cusp

Theorems & Definitions (55)

  • Theorem 1.1: Thurston
  • Corollary 1.2
  • Theorem 1.5: J2, Theorem 1.4
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • Corollary 1.11
  • Corollary 1.12
  • ...and 45 more