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Hyperbolic Handlebody Complements in 3-Manifolds

Colin Adams, Francisco Gomez-Paz, Jiachen Kang, Lukas Krause, Gregory Li, Chloe Marple, Ziwei Tan

TL;DR

The paper proves that any compact orientable 3-manifold $M$ (after capping spherical boundaries and removing torus boundaries) contains genus-$n$ handlebodies whose complements are tg-hyperbolic for any $n\ge 2$, extending Myers’ knot-exterior construction via knotted graph tangles to raise genus. It establishes incompressibility and simplicity of the relevant pieces through explicit fundamental-group calculations and a gluing lemma framework, enabling the assembled handlebody complements to be tg-hyperbolic by Thurston’s theory. The authors show this yields a volume spectrum $v_n^{min}(M)$ and, in a volume-bounds direction, extend the octahedral decomposition to generalized octahedra for spatial graphs in $S^3$ or thickened surfaces, giving explicit upper bounds proportional to crossing number and computing the maximal volume via the ideal, right-angled cuboctahedron. Collectively, the results provide both structural hyperbolicity inside arbitrary 3-manifolds and quantitative volume estimates tied to graph projections, with implications for hyperbolic invariants of non-hyperbolic hosts.

Abstract

Let $M_0$ be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold $M$ contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.

Hyperbolic Handlebody Complements in 3-Manifolds

TL;DR

The paper proves that any compact orientable 3-manifold (after capping spherical boundaries and removing torus boundaries) contains genus- handlebodies whose complements are tg-hyperbolic for any , extending Myers’ knot-exterior construction via knotted graph tangles to raise genus. It establishes incompressibility and simplicity of the relevant pieces through explicit fundamental-group calculations and a gluing lemma framework, enabling the assembled handlebody complements to be tg-hyperbolic by Thurston’s theory. The authors show this yields a volume spectrum and, in a volume-bounds direction, extend the octahedral decomposition to generalized octahedra for spatial graphs in or thickened surfaces, giving explicit upper bounds proportional to crossing number and computing the maximal volume via the ideal, right-angled cuboctahedron. Collectively, the results provide both structural hyperbolicity inside arbitrary 3-manifolds and quantitative volume estimates tied to graph projections, with implications for hyperbolic invariants of non-hyperbolic hosts.

Abstract

Let be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.

Paper Structure

This paper contains 5 sections, 21 theorems, 17 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental Group Calculations
  4. Proof of Main Theorem
  5. Volume bounds

Key Result

Theorem 1.1

Given a compact orientable 3-manifold $M$ with or without boundary, and a positive integer $n \geq 2$, $M$ contains a genus $n$ handlebody such that the closure of its complement is tg-hyperbolic.

Figures (23)

  • Figure 1: True Lovers Tangle $X$ with exterior $E(X)$ in a closed 3-ball
  • Figure 2: Knotted graph $J$ with exterior $E(J)$ in a closed 3-ball
  • Figure 3: The knotted graph $I$ with the same exterior as $J$
  • Figure 4: Showing the tangle exterior of $I$ and the spatial graph exterior of $J$ are equivalent
  • Figure 5: Decompose $E(I)$ into $P_1\cup P_2\cup P_3$ and $P$ with gluing faces $F_1,F_2,F_3$
  • ...and 18 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Thurston Thurston
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8: Gluing Lemma
  • ...and 39 more