Geometry of planar surfaces and exceptional fillings

Neil R. Hoffman, Jessica S. Purcell

Published 2015-04-07Version 1

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 a limit point of lengths of slopes for small Seifert fibered space fillings.