[Paper Review] The metric space of geodesic laminations on a surface II: small surfaces

Francis Bonahon, Xiaodong Zhu|arXiv (Cornell University)|Aug 28, 2003

Geometric Analysis and Curvature Flows3 references4 citations

TL;DR

This paper investigates the topology and metric geometry of geodesic lamination spaces on the once-punctured torus and the 4-times-punctured sphere, showing that their spaces of laminations embed into the circle as Cantor sets with isolated points. It establishes that, under a logarithmic transformation of the Hausdorff metric, these spaces have Hausdorff dimension 2 with zero 2-dimensional measure, revealing a rich fractal structure distinct from the 1-dimensional Thurston boundary.

ABSTRACT

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic laminations, when it is endowed with the natural metric which, for small distances, is -1 over the logarithm of the Hausdorff metric. The key ingredient is an estimate of the Hausdorff metric between two simple closed geodesics in terms of their respective slopes.

Motivation & Objective

• To determine the topological structure of the space of geodesic laminations on the once-punctured torus and the 4-times-punctured sphere.
• To compute the Hausdorff dimension of this space when equipped with a logarithmic transformation of the Hausdorff metric.
• To establish a natural embedding of the lamination space into the circle, linking components of the complement of a Cantor set to simple closed geodesics.
• To analyze the metric properties of the space under a new metric $d_{\log}$, which is invariant under Lipschitz equivalence and well-suited for dimension theory.
• To contrast the fractal geometry of the lamination space with the 1-dimensional Thurston boundary of measured laminations.

Proposed method

• The paper uses a topological decomposition of the lamination space into a Cantor set $K$ and a countable set of isolated points, based on the slope of simple closed geodesics.
• It establishes a one-to-one correspondence between components of $\mathbb{S}^1 - K$ and simple closed curves on the surface.
• A key technical tool is a quantitative estimate relating the Hausdorff distance between two simple closed geodesics to their respective slopes.
• The authors define a new metric $d_{\log}$, where $d_{\log}(x,y) = -1 / \log d_H(x,y)$ for small $d_H$, which is well-behaved under Lipschitz equivalence.
• They compute the Hausdorff dimension using a covering argument on the set of irrational slopes, leveraging Farey intervals and continued fraction expansions.
• The proof of zero 2-dimensional measure uses a covering construction based on Farey intervals with large ratios of denominators, showing that $\sum r_i^2$ can be made arbitrarily small.

Experimental results

Research questions

• RQ1 What is the topological structure of the space of geodesic laminations on the once-punctured torus and the 4-times-punctured sphere?
• RQ2 How does the Hausdorff dimension of the lamination space behave under a logarithmic transformation of the Hausdorff metric?
• RQ3 Can the space of laminations be naturally embedded into the circle in a way that reflects the geometry of simple closed curves?
• RQ4 What is the relationship between the number of isolated points in each component of $\mathbb{S}^1 - K$ and the topology of the surface?
• RQ5 How does the metric structure of the lamination space compare to the 1-dimensional Thurston boundary of measured laminations?

Key findings

• For the once-punctured torus, $\mathcal{L}_0(S)$ is homeomorphic to $K \cup L_3$, where $K$ is the Cantor set and $L_3$ consists of exactly three isolated points per component of $\mathbb{S}^1 - K$.
• For the 4-times-punctured sphere, $\mathcal{L}_0(S)$ is homeomorphic to $K \cup L_7$, with seven isolated points per component.
• The closure of the set of simple closed curves in each case is $K \cup L_1$, with exactly one isolated point per component.
• The Hausdorff dimension of $\left(\mathcal{L}_0(S), d_{\log}\right)$ is exactly 2 for both surfaces.
• The 2-dimensional Hausdorff measure of $\left(\mathcal{L}_0(S), d_{\log}\right)$ is zero, indicating that the space is 'thin' in 2D despite having dimension 2.
• The results contrast sharply with the 1-dimensional Thurston boundary $\mathcal{PML}(S)$, which is homeomorphic to the circle and has dimension 1.