[Paper Review] Determining hyperbolicity of compact orientable 3-manifolds with torus boundary

Haraway, Christian Y. Robert|arXiv (Cornell University)|Oct 27, 2014

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper presents an algorithm to determine whether a compact orientable 3-manifold with torus boundary admits a complete finite-volume hyperbolic metric, leveraging Thurston's hyperbolization theorem and normal surface theory. The key contribution is a decision procedure that reduces the conjecture of Gabai, Meyerhoff, and Milley to a computable problem.

ABSTRACT

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.

Motivation & Objective

• To develop a decision procedure for determining whether a compact orientable 3-manifold with torus boundary admits a finite-volume hyperbolic metric.
• To apply Thurston's hyperbolization theorem and normal surface theory to construct an effective algorithm.
• To reduce the conjecture of Gabai, Meyerhoff, and Milley to a computable problem.
• To provide a theoretical foundation for algorithmic verification of hyperbolic structures in 3-manifolds with torus boundary.

Proposed method

• Utilize Thurston's hyperbolization theorem for Haken manifolds as a theoretical basis for the existence of hyperbolic structures.
• Apply normal surface theory to analyze embedded surfaces and detect geometric structures.
• Construct an algorithmic framework that combines topological and geometric criteria to test for hyperbolicity.
• Use computational techniques to verify the existence of a complete finite-volume hyperbolic metric in the interior of the manifold.
• Reduce the problem to a finite computation by exploiting the finiteness properties of normal surfaces in triangulations.
• Leverage the structure of torus boundary components to constrain the possible geometric types of the manifold.

Experimental results

Research questions

• RQ1 Can an algorithm be constructed to determine whether a compact orientable 3-manifold with torus boundary admits a complete finite-volume hyperbolic metric?
• RQ2 How can Thurston's hyperbolization theorem be made algorithmically effective for such manifolds?
• RQ3 What role does normal surface theory play in detecting hyperbolic structures in 3-manifolds with torus boundary?
• RQ4 To what extent can the conjecture of Gabai, Meyerhoff, and Milley be reduced to a computable decision problem?
• RQ5 What are the necessary and sufficient conditions for hyperbolicity in this class of 3-manifolds, expressible via algorithmic means?

Key findings

• The paper establishes a complete algorithmic procedure to determine whether a compact orientable 3-manifold with torus boundary admits a finite-volume hyperbolic metric.
• The algorithm is derived from Thurston's hyperbolization theorem and normal surface theory, ensuring theoretical soundness.
• The existence of a hyperbolic structure is decidable within this framework, providing a constructive method.
• The conjecture of Gabai, Meyerhoff, and Milley is reduced to a finite computation, making it amenable to algorithmic verification.
• The method provides a systematic way to analyze the geometric structure of 3-manifolds with torus boundary using topological and geometric tools.