[Paper Review] Minimum volume cusped hyperbolic three-manifolds

David Gabai, Robert Meyerhoff+1 more|ArXiv.org|May 30, 2007

Geometric and Algebraic Topology7 references4 citations

TL;DR

This paper identifies all one-cusped orientable hyperbolic 3-manifolds with volume ≤ 2.848 by proving they arise from Dehn filling on 21 specific cusped hyperbolic manifolds, using Mom-structure technology and rigorous computational verification. The key result is that only 10 such manifolds (m003 to m017 in the SnapPea census) satisfy the volume bound, confirming the Weeks manifold as the unique smallest-volume closed hyperbolic 3-manifold.

ABSTRACT

This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic three-manifolds with volume <= 2.848 and all closed hyperbolic three-manifolds with volume <= 0.943. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperbolic 3-manifold.

Motivation & Objective

• To classify all one-cusped orientable hyperbolic 3-manifolds with volume ≤ 2.848.
• To extend the Mom-structure theory to prove that such manifolds arise from Dehn filling on specific minimal-volume cusped manifolds.
• To confirm the Weeks manifold as the unique smallest-volume closed hyperbolic 3-manifold using volume bounds and Dehn filling analysis.
• To provide a rigorous computational framework for verifying geometric and topological properties of low-volume hyperbolic 3-manifolds.

Proposed method

• Utilizes the Mom-structure framework from [GMM2], embedding geometrically defined geodesic arcs and hexagons in the hyperbolic 3-space to detect internal topological structures.
• Applies rigorous floating-point computations to verify geometric properties of horoball packings and overlap areas in the universal cover.
• Employs the $\log(3)/2$ theorem and tube radius bounds to constrain possible fillings and volume growth under Dehn surgery.
• Uses the $\operatorname{lessvol}$ and $\operatorname{overlapArea}$ functions to analyze volume reduction and geometric consistency in Dehn filling operations.
• Analyzes the Dehn surgery spaces of 21 candidate manifolds to exhaustively list all resulting one-cusped manifolds with volume ≤ 2.848.
• Applies Agol's formula relating volume of closed manifolds to drilled manifolds and tube radii to link closed and cusped cases.

Experimental results

Research questions

• RQ1 Which one-cusped orientable hyperbolic 3-manifolds have volume ≤ 2.848, and how can they be systematically generated via Dehn filling?
• RQ2 Can the Mom-structure technology be used to rigorously classify low-volume cusped hyperbolic 3-manifolds?
• RQ3 Is the Weeks manifold the unique smallest-volume closed hyperbolic 3-manifold, and can this be proven via Dehn filling on low-volume cusped manifolds?
• RQ4 What is the minimal volume threshold beyond which the current Mom-structure classification fails, and how can it be extended?

Key findings

• The only one-cusped orientable hyperbolic 3-manifolds with volume ≤ 2.848 are m003, m004, m006, m007, m009, m010, m011, m015, m016, and m017 from the SnapPea census.
• All such manifolds arise by Dehn filling on one of 21 specific cusped hyperbolic 3-manifolds listed in Figure 1, which are the Mom-2 and Mom-3 manifolds from [GMM2].
• The Weeks manifold is the unique closed orientable hyperbolic 3-manifold of smallest volume, obtained by (5,1) or (5,2) filling on the Whitehead link complement.
• The volume bound of 2.848 is sufficient to classify all such one-cusped manifolds, and this bound is tight given the next non-Mom-3 manifold (m069) has volume > 3.4.
• The analysis confirms that the Weeks manifold has volume less than 2.848/3.02, and any closed hyperbolic 3-manifold with smaller volume must arise from Dehn filling on one of the 10 listed one-cusped manifolds.
• The results suggest that extending the volume bound to ~3.7 could yield the first infinite sequence of volumes limiting on the Whitehead link complement.