Geometry of planar surfaces and exceptional fillings

Neil R. Hoffman, Jessica S. Purcell

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)185-201
Number of pages17
JournalBulletin of the London Mathematical Society
Issue number2
StatePublished - Apr 1 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 London Mathematical Society.


  • 57M27 (secondary)
  • 57M50 (primary)


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