About: 2π theorem

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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 or

Attributes	Values
rdf:type	yago:WikicatTheoremsInGeometry yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	2π theorem (en)
rdfs: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. 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 or (en)
dcterms:subject	3-manifolds Theorems in geometry
Wikipage page ID	11945645 (xsd:integer)
Wikipage revision ID	999895412 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime decomposition (3-manifold) Inventiones Mathematicae 3-manifolds Mathematics Geometry & Topology Geometrization conjecture Fundamental group William Thurston Theorems in geometry Atoroidal Hyperbolic 3-manifold Colin Adams (mathematician) Topology (journal) Mikhail Leonidovich Gromov Seifert fiber space Word hyperbolic group Dehn filling Horoball
sameAs	2π theorem 2π theorem 2π theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Math dbt:Mvar dbt:Short_description dbt:Harvs
authorlink	Ian Agol (en) Marc Lackenby (en)
first	Ian (en) Marc (en)
last	Agol (en) Lackenby (en)
year	2000 (xsd:integer)
has 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 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 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 and Marc Lackenby 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. (en)
prov:wasDerivedFrom	wikipedia-en:2π_theorem?oldid=999895412&ns=0
page length (characters) of wiki page	3663 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:2π_theorem
is Link from a Wikipage to another Wikipage of	Gromov-Thurston 2π theorem 2p theorem Marc Lackenby Mikhael Gromov (mathematician) List of theorems List of topics related to π 6 theorem Gromov-Thurston 2pi theorem Gromov–Thurston 2pi theorem Gromov–Thurston 2π theorem 2pi theorem
is Wikipage redirect of	Gromov-Thurston 2π theorem 2p theorem 6 theorem Gromov-Thurston 2pi theorem Gromov–Thurston 2pi theorem Gromov–Thurston 2π theorem 2pi theorem
is known for of	Mikhael Gromov (mathematician)
is foaf:primaryTopic of	wikipedia-en:2π_theorem

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