[Paper Review] The geometry of the disk complex

Howard Masur, Saul Schleimer|Warwick Research Archive Portal (University of Warwick)|Oct 15, 2010

Geometric and Algebraic Topology31 references4 citations

TL;DR

This paper establishes that the disk complex of a handlebody is Gromov hyperbolic by classifying topological obstructions—called 'holes'—that cause distortion between the disk complex and the curve complex. Using a novel distance estimate based on subsurface projections and techniques from Teichmüller theory and train track splitting sequences, the authors develop a coarse algorithm to compute Hempel distance within a genus-dependent error bound, resolving a key geometric problem in 3-manifold topology.

ABSTRACT

We give a distance estimate for the metric on the disk complex and show that it is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus.

Motivation & Objective

• To understand the geometric distortion between the disk complex and the curve complex in handlebody boundaries.
• To classify the topological obstructions—'holes'—that prevent the disk complex from being quasi-isometrically embedded in the curve complex.
• To establish a coarse algorithm for computing the Hempel distance of a Heegaard splitting, with error bounded only by the genus.
• To prove the disk complex is Gromov hyperbolic using a new axiomatic framework based on subsurface projections and marking paths.

Proposed method

• Classify holes in the disk complex as non-annular, compressible, or incompressible, linking incompressible holes to I-bundles over surfaces.
• Use subsurface projection distances to define a coarse distance estimate in the disk complex, coarsely equal to the sum of projections over all holes.
• Apply the Teichmüller geodesic machine for non-orientable surfaces and the train track splitting sequence machine for the disk complex to construct marking paths.
• Leverage the hyperbolicity of the curve complex and the quasi-convexity of the disk set within the curve complex as foundational inputs.
• Construct an algorithm that computes a disk close to the image of a marking under the disk complex inclusion, using bounded error terms derived from geometric constants.
• Use shortcut arguments in the curve complex to bound distances between algorithm outputs and target markings, ensuring bounded error.

Experimental results

Research questions

• RQ1 What topological structures (holes) obstruct the disk complex from being quasi-isometrically embedded in the curve complex?
• RQ2 Can the distance in the disk complex be coarsely estimated using subsurface projections, and if so, how?
• RQ3 Is the disk complex Gromov hyperbolic, and what geometric machinery supports this result?
• RQ4 Can the Hempel distance of a Heegaard splitting be computed coarsely, with error depending only on genus?
• RQ5 How do the geometric properties of the disk complex relate to the structure of I-bundles and arc complexes on non-orientable surfaces?

Key findings

• The disk complex is Gromov hyperbolic, as proven via a new axiomatic framework based on subsurface projections and marking paths.
• Holes in the disk complex are classified: non-annular, compressible (with filling boundary disks), or incompressible (arising from I-bundles over surfaces, possibly non-orientable).
• A coarse algorithm computes the Hempel distance of a Heegaard splitting within an additive error bounded by a constant $ K(g) $ depending only on the genus $ g $.
• The distance in the disk complex is coarsely equal to the sum of subsurface projection distances over all holes, establishing a key distance estimate.
• The proof avoids the hierarchy machine, instead using the rigid Teichmüller geodesic machine and flexible train track splitting sequences for different complexes.
• The additive error in the algorithm is bounded by a universal constant depending only on genus, derived from geometric constants like hyperbolicity and quasi-convexity parameters.