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.

摘要

上同调分形是与双曲三流形中的上同调类自然关联的图像。我们通过光线追踪到固定的视觉半径，实时为尖的、不完整的和闭合的双曲三流形生成这些图像。我们在尝试不使用矢量图形来说明 Cannon-Thurston 映射时发现了上同调分形；当上同调类与纤维化对偶时，我们证明了这两者之间的对应关系。这使我们能够通过将我们的上同调分形图像与现有的 Cannon-Thurston 映射图片进行比较来验证我们的实现。

在一系列实验中，我们探索了上同调分形在视觉半径增加时的极限行为。受这些实验的启发，我们证明了上同调分形的值服从正态分布，但标准差不同。事实上，上同调分形不会收敛于极限函数。相反，我们表明极限是无穷远球体上的分布，仅取决于流形和上同调类。