Thurston hyperbolization theorem

E898489

mathematical theorem result in 3-manifold topology

The Thurston hyperbolization theorem is a fundamental result in 3-manifold topology that characterizes when certain 3-manifolds admit complete hyperbolic structures, forming a cornerstone of Thurston’s geometrization program.

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All labels observed (2)

Label	Occurrences
Thurston hyperbolic Dehn surgery theorem	1
Thurston hyperbolization theorem canonical	1

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in 3-manifold topology ⓘ
appliesTo	Haken 3-manifolds under suitable conditions ⓘ atoroidal Haken 3-manifolds with infinite fundamental group ⓘ many knot complements in S^3 ⓘ
assumes	irreducible 3-manifolds ⓘ sufficiently large 3-manifolds in the Haken setting ⓘ
author	William P. Thurston NERFINISHED ⓘ
characterizes	when certain 3-manifolds admit complete hyperbolic metrics ⓘ
concerns	3-manifolds ⓘ complete hyperbolic structures ⓘ geometric structures on 3-manifolds ⓘ hyperbolic 3-manifolds ⓘ
concludes	3-manifold admits a complete finite-volume hyperbolic metric under its hypotheses ⓘ 3-manifold is hyperbolic in the sense of Thurston’s geometrization ⓘ
field	3-manifold theory ⓘ geometric topology ⓘ hyperbolic geometry ⓘ
formalizes	conditions under which a 3-manifold supports a hyperbolic geometry ⓘ
hasConsequence	classification of many 3-manifolds as hyperbolic ⓘ existence of large families of hyperbolic 3-manifolds ⓘ
hasVersion	hyperbolization theorem for Haken 3-manifolds NERFINISHED ⓘ hyperbolization theorem for atoroidal Haken manifolds ⓘ
helpsProve	hyperbolicity of complements of many knots and links ⓘ
historicalPeriod	late 20th century ⓘ
implies	existence of hyperbolic structures on many Haken 3-manifolds ⓘ many 3-manifolds have unique hyperbolic structures up to isometry ⓘ
importance	cornerstone of modern 3-manifold topology ⓘ key component of the proof of geometrization for Haken manifolds ⓘ
influenced	Perelman’s work on geometrization ⓘ
inspired	subsequent generalizations of hyperbolization to other settings ⓘ
isCornerstoneOf	Thurston’s theory of 3-dimensional geometries NERFINISHED ⓘ
namedAfter	William P. Thurston NERFINISHED ⓘ
partOf	Thurston’s geometrization program NERFINISHED ⓘ
provedUsing	methods of low-dimensional topology and hyperbolic geometry ⓘ
relatedTo	Haken’s work on 3-manifolds ⓘ JSJ decomposition of 3-manifolds ⓘ Mostow–Prasad rigidity theorem NERFINISHED ⓘ Thurston geometrization conjecture NERFINISHED ⓘ hyperbolic Dehn surgery theorem NERFINISHED ⓘ prime decomposition of 3-manifolds ⓘ
status	proved ⓘ
uses	Haken hierarchy NERFINISHED ⓘ Kleinian group theory NERFINISHED ⓘ Mostow rigidity NERFINISHED ⓘ character variety methods ⓘ hyperbolic Dehn surgery techniques ⓘ incompressible surfaces ⓘ pleated surfaces ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kleinian group → relatedTo → Thurston hyperbolization theorem ⓘ

Dehn surgery → relatedTheorem → Thurston hyperbolization theorem ⓘ

this entity surface form: Thurston hyperbolic Dehn surgery theorem