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- 2π_theorem abstract "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. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics. A slope, i.e. unoriented isotopy class of simple closed curves on these boundaries, thus has a well-defined length by taking the minimal Euclidean length over all curves in the isotopy class. The 2π theorem states: a Dehn filling of M with each filling slope greater than 2π results in a 3-manifold with a complete metric of negative sectional curvature. In fact, this metric can be selected to be identical to the original hyperbolic metric outside the horoball neighborhoods. The basic idea of the proof is to explicitly construct a negatively curved metric inside each horoball neighborhood that matches the metric near the horospherical boundary. This construction, using cylindrical coordinates, works when the filling slope is greater than 2π. See Bleiler & Hodgson (1996) for complete details.According to the geometrization conjecture, these negatively curved 3-manifolds must actually admit a complete hyperbolic metric. A horoball packing argument due to Thurston shows that there are at most 48 slopes to avoid on each cusp to get a nonhyperbolic 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. Yet again assuming the geometrization conjecture, these manifolds have a complete hyperbolic metric. An argument of Agol's shows that there are at most 12 exceptional slopes.".
- 2π_theorem wikiPageID "11945645".
- 2π_theorem wikiPageLength "3356".
- 2π_theorem wikiPageOutDegree "19".
- 2π_theorem wikiPageRevisionID "664508992".
- 2π_theorem wikiPageWikiLink Atoroidal.
- 2π_theorem wikiPageWikiLink Category:3-manifolds.
- 2π_theorem wikiPageWikiLink Category:Theorems_in_geometry.
- 2π_theorem wikiPageWikiLink Colin_Adams_(mathematician).
- 2π_theorem wikiPageWikiLink Dehn_filling.
- 2π_theorem wikiPageWikiLink Fundamental_group.
- 2π_theorem wikiPageWikiLink Geometrization_conjecture.
- 2π_theorem wikiPageWikiLink Geometry_&_Topology.
- 2π_theorem wikiPageWikiLink Horoball.
- 2π_theorem wikiPageWikiLink Horosphere.
- 2π_theorem wikiPageWikiLink Hyperbolic_3-manifold.
- 2π_theorem wikiPageWikiLink Hyperbolic_Dehn_surgery.
- 2π_theorem wikiPageWikiLink Hyperbolic_group.
- 2π_theorem wikiPageWikiLink Inventiones_Mathematicae.
- 2π_theorem wikiPageWikiLink Mathematics.
- 2π_theorem wikiPageWikiLink Mikhail_Leonidovich_Gromov.
- 2π_theorem wikiPageWikiLink Prime_decomposition_(3-manifold).
- 2π_theorem wikiPageWikiLink Seifert_fiber_space.
- 2π_theorem wikiPageWikiLink Topology_(journal).
- 2π_theorem wikiPageWikiLink William_Thurston.
- 2π_theorem wikiPageWikiLink Word_hyperbolic_group.
- 2π_theorem wikiPageWikiLinkText "2 theorem".
- 2π_theorem wikiPageWikiLinkText "2π theorem".
- 2π_theorem authorlink "Ian Agol".
- 2π_theorem first "Ian".
- 2π_theorem hasPhotoCollection 2π_theorem.
- 2π_theorem last "Agol".
- 2π_theorem wikiPageUsesTemplate Template:Citation.
- 2π_theorem wikiPageUsesTemplate Template:Harvs.
- 2π_theorem wikiPageUsesTemplate Template:Harvtxt.
- 2π_theorem wikiPageUsesTemplate Template:Pi.
- 2π_theorem year "2000".
- 2π_theorem subject Category:3-manifolds.
- 2π_theorem subject Category:Theorems_in_geometry.
- 2π_theorem comment "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. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics. A slope, i.e.".
- 2π_theorem label "2π theorem".
- 2π_theorem sameAs m.02rz2yy.
- 2π_theorem sameAs Q4634017.
- 2π_theorem sameAs Q4634017.
- 2π_theorem wasDerivedFrom 2π_theorem?oldid=664508992.
- 2π_theorem isPrimaryTopicOf 2π_theorem.