A Stronger Gordon Conjecture and an Analysis of Free Bicuspid Manifolds with Small Cusps

Abstract

Thurston showed that for all but a finite number of Dehn Surgeries on a cusped hyperbolic 3-manifold, the resulting manifold admits a hyperbolic structure. Global bounds on this number have been set, and gradually improved upon, by a number of Mathematicians until Lackenby and Meyerhoff proved the sharp bound of 10, which is realized by the figure-eight knot exterior. We improve this result by proving a stronger version of Gordon’s conjecture: that excluding the figure-eight knot exterior, cusped hyperbolic 3-manifolds have at most 8 non-hyperbolic Dehn Surgeries. To do so we make use of the work of Gabai et. al. from a forthcoming paper which parameterizes measurements of the cusp, then uses a rigorous computer aided search of the space to classify all hyperbolic 3-manifolds up to a specified cusp size. Their approach hinges on the discreteness of manifold points in the parameter space, an assumption which cannot be made if the manifolds have infinite volume. In this paper we also show that infinite-volume manifolds, which must be Free Bicuspid, can have cusp volume as low as 3.159. As such, these manifolds are a concern for any future expansion of the approach of Gabai et. al.