Hyperbolic 3-manifolds of low cusp volume

TL;DR

This paper classifies hyperbolic 3-manifolds with small cusp volume, identifying unique minimal volume manifolds and extending classification results for low volume hyperbolic 3-manifolds.

Contribution

It provides a classification of hyperbolic 3-manifolds with maximal cusp volume at most 2.62 and identifies unique minimal volume manifolds among them.

Findings

Figure-8 knot complement is the only 1-cusped hyperbolic manifold with nine or more non-hyperbolic fillings.

Figure-8 knot complement and its sister are the unique hyperbolic manifolds with minimal volume maximal cusps.

Extended classification results for low volume closed and cusped hyperbolic 3-manifolds.

Abstract

We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.