Hyperbolic 3-manifolds with low cusp volume

Andrew Yarmola (Princeton)

Abstract : For a cusped hyperbolic 3-manifold, one can consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present a classification of the infinite family of hyperbolic 3-manifolds of cusp volume < 2.62 and the implications of this classification. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings, allowing one to show that the figure eight knot complement is the only 1-cusped hyperbolic 3-manifold with > 8 exceptional slopes. Our work also allows us to classify the first 3 smallest volume closed hyperbolic manifolds. As in some other results on hyperbolic 3-manifolds of low volume, our technique utilizes a rigorous computer assisted search. This talk will focus on providing sufficient background to explain our approach and describe our conclusions. This work is joint with David Gabai, Robert Meyerhoff, Nathaniel Thurston, and Robert Haraway.