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Expansion joints in hyperbolic manifolds

Alex Elzenaar

TL;DR

The paper develops a constructive theory of deforming hyperbolic 3-manifolds via cone-manifold structures, using explicit polyhedral (octahedral) decompositions to realize expansion joints that localize deformations near cusps. It extends Thurston’s Dehn-filling perspective to rank-1 cusp creation by cone angles around ideal arcs, producing a continuous path of holonomies from complete manifolds (like Borromean rings complements) to lantern-type limits, and shows how Dehn fillings and cone angles control cusp shapes and hyperbolicity. A central novelty is the introduction of concave lenses as expansion joints, providing a general mechanism to deform a manifold while preserving the rest of its hyperbolic geometry, with plentiful examples from stacked Borromean rings and fully augmented links. These methods yield concrete results on unknotting tunnels for highly twisted 2-bridge links and establish broad, actionable connections between finite-volume and infinite-covolume Kleinian groups within a polyhedral, constructive framework.

Abstract

Deformations of hyperbolic manifolds through metrics with cone singularities along closed loops were first studied by Thurston as continuous realisations of Dehn fillings. Instead of gluing singular solid tori into rank $2$ cusps, we glue singular $2$-handles into rank $1$ cusps. Our method is to find substructures within which the hyperbolic metric can be `fractured' in a controlled way by direct manipulation of a fundamental polyhedron, changing the cone angle around an ideal arc to interpolate between cusped hyperbolic manifolds and hyperbolic manifolds with conformal surfaces on the visual boundary. As an application, we use cone deformations of a family of arithmetic manifolds derived from the Borromean rings to show that the upper unknotting tunnels of highly twisted $2$-bridge links can be drilled out by cone deformations. We also show that our structures arise naturally in fully augmented links, providing a large family of examples.

Expansion joints in hyperbolic manifolds

TL;DR

The paper develops a constructive theory of deforming hyperbolic 3-manifolds via cone-manifold structures, using explicit polyhedral (octahedral) decompositions to realize expansion joints that localize deformations near cusps. It extends Thurston’s Dehn-filling perspective to rank-1 cusp creation by cone angles around ideal arcs, producing a continuous path of holonomies from complete manifolds (like Borromean rings complements) to lantern-type limits, and shows how Dehn fillings and cone angles control cusp shapes and hyperbolicity. A central novelty is the introduction of concave lenses as expansion joints, providing a general mechanism to deform a manifold while preserving the rest of its hyperbolic geometry, with plentiful examples from stacked Borromean rings and fully augmented links. These methods yield concrete results on unknotting tunnels for highly twisted 2-bridge links and establish broad, actionable connections between finite-volume and infinite-covolume Kleinian groups within a polyhedral, constructive framework.

Abstract

Deformations of hyperbolic manifolds through metrics with cone singularities along closed loops were first studied by Thurston as continuous realisations of Dehn fillings. Instead of gluing singular solid tori into rank cusps, we glue singular -handles into rank cusps. Our method is to find substructures within which the hyperbolic metric can be `fractured' in a controlled way by direct manipulation of a fundamental polyhedron, changing the cone angle around an ideal arc to interpolate between cusped hyperbolic manifolds and hyperbolic manifolds with conformal surfaces on the visual boundary. As an application, we use cone deformations of a family of arithmetic manifolds derived from the Borromean rings to show that the upper unknotting tunnels of highly twisted -bridge links can be drilled out by cone deformations. We also show that our structures arise naturally in fully augmented links, providing a large family of examples.

Paper Structure

This paper contains 9 sections, 10 theorems, 6 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The classical Borromean rings
  3. Stacked Borromean rings and lantern manifolds
  4. Hyperbolic structure
  5. Dehn filling
  6. Concave lenses and other expansion joints
  7. Fully augmented links
  8. Further examples
  9. The polyhedron theorem for cone manifolds

Key Result

Proposition 2.1

There are exactly $16$ solutions in $\mathbb{C}^9$ to the polynomial system just described which satisfy the non-degeneracy conditions which prevent various parabolics from degenerating to the identity. Eight of these give groups conjugate after change of generators to the group $G_{2\pi}$ with parameters The remaining eight give groups conjugate after change of generators to the group $G_0$ wit

Figures (20)

  • Figure 1: The $n$-stacked Borromean rings for small $n$.
  • Figure 2: The lantern manifold is produced by deforming the cone angle in the Borromean rings complement around the dotted arc $\ast$ from $2\pi$ to $0$.
  • Figure 3: The combinatorics and geometry of $B^3$.
  • Figure 4: The lantern manifold $M$: a genus $2$ handlebody with two drilled loops. Here, and elsewhere in the paper, manifolds are drawn from the perspective of a viewer in the interior.
  • Figure 5: Data associated with the Borromean rings group $G_{2\pi}$.
  • ...and 15 more figures

Theorems & Definitions (32)

  • proof
  • Example 2.2: The Borromean rings
  • Example 2.3: The lantern group
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 22 more