Geometry of planar surfaces and exceptional fillings

TL;DR

This paper explores the maximum length of slopes in hyperbolic 3-manifolds for various exceptional Dehn fillings, revealing new bounds and connections to horoball packings.

Contribution

It constructs hyperbolic 3-manifolds with the longest known slopes for reducible fillings and links the problem to horoball packings on planar surfaces.

Findings

Longest slopes for reducible fillings are identified.

Six is not the maximum slope length for small Seifert fibered space fillings.

The problem relates to maximal density horoball packings.

Abstract

If a hyperbolic 3-manifold admits an exceptional Dehn filling, then the length of the slope of that Dehn filling is known to be at most six. However, the bound of six appears to be sharp only in the toroidal case. In this paper, we investigate slope lengths of other exceptional fillings. We construct hyperbolic 3-manifolds that have the longest known slopes for reducible fillings. As an intermediate step, we show that the problem of finding the longest such slope is equivalent to a problem on the maximal density horoball packings of planar surfaces, which should be of independent interest. We also discuss lengths of slopes of other exceptional Dehn fillings, and prove that six is not realized by a slope corresponding to a small Seifert fibered space filling.