2π theorem

In mathematics, the 2π theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold.

A horoball packing argument due to Thurston shows that there are at most 48 slopes to avoid on each cusp to get a hyperbolic 3-manifold.

For one-cusped hyperbolic 3-manifolds, an improvement due to Colin Adams gives 24 exceptional slopes.

This result was later improved independently by Ian Agol (2000) and Marc Lackenby (2000) with the 6 theorem.

The "6 theorem" states that Dehn filling along slopes of length greater than 6 results in a hyperbolike 3-manifold, i.e. an irreducible, atoroidal, non-Seifert-fibered 3-manifold with infinite word hyperbolic fundamental group.