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Cohomology fractals, Cannon-Thurston maps, and the geodesic flow

David Bachman, Matthias Goerner, Saul Schleimer, Henry Segerman

TL;DR

This work addresses visualizing and understanding Cannon–Thurston maps via cohomology fractals on hyperbolic 3-manifolds. It develops rigorous, equivalent definitions of the cohomology fractal $\Phi_R$, proves a central limit theorem for its values along geodesic flow, and introduces a pixel theorem that describes the limiting distribution at infinity. The authors provide a real-time ray-tracing implementation, relate fractals to CT maps in fibred cases, and analyze incomplete and closed manifolds, Dehn surgery, and numerical stability. The results yield a distributional perspective on boundary maps in hyperbolic geometry, enabling robust visualization and deeper connections to dynamical systems and Thurston theory.

Abstract

Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class.

Cohomology fractals, Cannon-Thurston maps, and the geodesic flow

TL;DR

This work addresses visualizing and understanding Cannon–Thurston maps via cohomology fractals on hyperbolic 3-manifolds. It develops rigorous, equivalent definitions of the cohomology fractal , proves a central limit theorem for its values along geodesic flow, and introduces a pixel theorem that describes the limiting distribution at infinity. The authors provide a real-time ray-tracing implementation, relate fractals to CT maps in fibred cases, and analyze incomplete and closed manifolds, Dehn surgery, and numerical stability. The results yield a distributional perspective on boundary maps in hyperbolic geometry, enabling robust visualization and deeper connections to dynamical systems and Thurston theory.

Abstract

Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class.

Paper Structure

This paper contains 48 sections, 22 theorems, 92 equations, 24 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Triangulations
  3. Combinatorics
  4. Geometry
  5. Ideal triangulations
  6. Material triangulations
  7. Cannon--Thurston maps
  8. Illustrating Cannon--Thurston maps
  9. Vector and raster graphics
  10. Motivating raster graphics
  11. Cohomology fractals
  12. Matching figures
  13. Lightning curves
  14. Implementation
  15. Ray-tracing
  16. ...and 33 more sections

Key Result

Theorem 3.1

Suppose $M$ is a connected, oriented, finite volume hyperbolic three-manifold. Suppose that $\alpha \colon F \to M$ is a fibre of a surface bundle structure on $M$. Then there is an extension of ${\widetilde{\alpha}}$ to a continuous and equivariant (with respect to the fundamental group of $M$) map

Figures (24)

  • Figure 1.1: Matching up Cannon--Thurston map images with the cohomology fractal for m004, the figure-eight knot complement. Compare our Figure \ref{['Fig:CTPixelBW']} with Figure 10.11 of Indra's PearlsIndrasPearls, which was produced by paint-filling a vector graphics image Wright19.
  • Figure 2.1: The figure-eight knot complement. This manifold is known as m004 in the SnapPy census. The black lines in Figure \ref{['Fig:FigureEightKnot']} cut the tube into triangles, corresponding to the eight green triangles in Figure \ref{['Fig:FigureEightTriangulation']}.
  • Figure 3.1: The various spaces and maps involved in constructing the Cannon--Thurston map $\Psi$.
  • Figure 4.1: Views in the universal cover of the figure-eight knot complement.
  • Figure 5.1: Cohomology fractals for m004, with various values of $R$.
  • ...and 19 more figures

Theorems & Definitions (97)

  • Example 2.1
  • Theorem 3.1
  • Remark 4.2
  • Definition 5.1
  • Definition 5.2
  • Definition 5.3
  • Definition 5.4
  • Definition 5.5
  • Remark 5.6
  • Remark 5.7
  • ...and 87 more