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BoGwang Jeon

Publications and source records attributed to BoGwang Jeon.

13 recordsLinked to original sources
arXiv2025

Classification of Hyperbolic Dehn fillings II: Quadratic case

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.

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arXiv2023

Hyperbolic Dehn filling, volume, and transcendentality

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.

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arXiv2022

Classification of hyperbolic Dehn fillings I

Let $M$ be a $2$-cusped hyperbolic $3$-manifold. By the work of Thurston, the product of the derivatives of the holonomies of core geodesics of each Dehn filling of $M$ is an invariant of it. In this paper, we classify Dehn fillings of $M$ with sufficiently large coefficients using this invariant. Further, for any given two Dehn fillings of $M$ (with sufficiently larger coefficients), if their aforementioned invariants are the same, it is shown their complex volumes are the same as well.

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arXiv2021

On the trace fields of hyperbolic Dehn fillings

Assuming Lehmer's conjecture, we estimate the degree of the trace field $K(M_{p/q})$ of a hyperbolic Dehn-filling $M_{p/q}$ of a 1-cusped hyperbolic 3-manifold $M$ by $$ \dfrac{1}{C}(\max\;\{|p|,|q|\})\leq \text{deg }K(M_{p/q}) \leq C(\max\;\{|p|,|q|\}) $$ where $C=C_M$ is a constant that depends on $M$.

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arXiv2020

On anomalous subvarieties of holonomy varieties of hyperbolic 3-manifolds

The goal of this paper is to explore the interplay between two seemingly distinct fields. More precisely, let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold with rationally independent cusp shapes, and $\mathcal{X}$ be its holonomy variety. We study the structure of anomalous subvarieties of $\mathcal{X}$, a concept originating in arithmetic geometry, and relate it to various geometric properties of $\mathcal{M}$. First, we show that every maximal anomalous subvariety of $\mathcal{X}$ containing the identity is its subvariety of codimension $1$ which arises by keeping one cusp of $\mathcal{M}$ complete. Second, we show that, if $\mathcal{X}$ is degenerated by its anomalous subvarieties (i.e., $\mathcal{X}^{oa}=\emptyset$), then $\mathcal{M}$ has cusps which are, while keeping some other cusps of it complete, strongly geometrically isolated from the rest. Finally, we completely classify and characterize the case $\mathcal{X}^{oa}=\emptyset$ for the holonomy variety $\mathcal{X}$ of any $2$-cusped hyperbolic $3$-manifold.

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arXiv2019

Rigidity in hyperbolic Dehn filling

This paper concerns with a rigidity of core geodesics in hyperbolic Dehn fillings. For instance, for an $n$-cusped hyperbolic $3$-manifold $M$ having non-symmetric cusp shapes, we show any Dehn filling of $M$ with sufficiently large coefficient is uniquely determined by the product of the holonomies of its core geodesics. We also explore various implications of the main results. An appendix by I. Agol provides an alternative geometric proof of one of the corollaries of our main arguments.

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arXiv2019

The Unlikely Intersection Theory and the Cosmetic Surgery Conjecture

Update: The Cosmetic Surgery Conjecture modulo finitely many Dehn-filling coefficients has been a well-known classical result, so the first main result of this paper is not new. (But the author was initially unaware of this fact, and the tools and techniques used here are very different from all the classically known methods.) The second main result of the paper, that is, the generalized Cosmetic Surgery Conjecture for the 2-cusped case is new, but superseded by the author's later work.

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arXiv2017

Realizing algebraic invariants of hyperbolic surfaces

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.

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arXiv2013

Hyperbolic three manifolds of bounded volume and trace field degree

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a manifold has linearly independent cusp shapes, we show that the manifold has the desired property.To prove the results, we use the proof of the Bounded Height Conjecture in arithmetic geometry.

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arXiv2012

Heegaard genera in congruence towers of hyperbolic 3-manifolds

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $ε>0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x) > (\text{vol}(M_i))^{\tfrac{1}{4}-ε}$. We get similar results in the arithmetic non-compact case.

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