Mostow rigidity theorem

E898487

mathematical theorem result in differential geometry result in geometric topology rigidity theorem

The Mostow rigidity theorem is a fundamental result in geometry and topology stating that, in dimensions greater than two, the large-scale geometry of a complete finite-volume hyperbolic manifold is uniquely determined by its fundamental group, implying strong rigidity for such structures.

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Observed surface forms (1)

Surface form	Occurrences
Mostow rigidity	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in differential geometry ⓘ result in geometric topology ⓘ rigidity theorem ⓘ
alsoKnownAs	Mostow strong rigidity theorem NERFINISHED ⓘ
appliesTo	complete finite-volume hyperbolic manifolds ⓘ hyperbolic manifolds of dimension at least 3 ⓘ
category	theorems in geometric topology ⓘ theorems in hyperbolic geometry ⓘ
concerns	discrete subgroups of Lie groups ⓘ isomorphisms of fundamental groups ⓘ lattices in rank-one Lie groups ⓘ
consequence	homeomorphic finite-volume hyperbolic manifolds of dimension at least 3 are isometric ⓘ outer automorphism group of the fundamental group of a closed hyperbolic n-manifold (n>=3) is finite ⓘ
dimensionCondition	>= 3 ⓘ
doesNotApplyTo	hyperbolic surfaces of dimension 2 ⓘ
field	Riemannian geometry NERFINISHED ⓘ geometric group theory ⓘ geometry ⓘ hyperbolic geometry ⓘ topology ⓘ
generalizationOf	rigidity results for lattices in higher-rank Lie groups ⓘ
hasGeneralization	Margulis superrigidity theorem NERFINISHED ⓘ Prasad rigidity results NERFINISHED ⓘ
historicalPeriod	20th-century mathematics ⓘ
holdsFor	closed hyperbolic manifolds of dimension at least 3 ⓘ finite-volume non-compact hyperbolic manifolds of dimension at least 3 ⓘ
implies	strong rigidity of hyperbolic structures in dimension at least 3 ⓘ topological equivalence implies geometric equivalence for finite-volume hyperbolic manifolds of dimension at least 3 ⓘ uniqueness of hyperbolic metric up to isometry for a given fundamental group in dimension at least 3 ⓘ
mainStatement	any isomorphism between the fundamental groups of two complete finite-volume hyperbolic manifolds of dimension at least 3 is induced by a unique isometry between the manifolds ⓘ the large-scale geometry of a complete finite-volume hyperbolic manifold of dimension at least 3 is determined by its fundamental group ⓘ
namedAfter	George Daniel Mostow NERFINISHED ⓘ
provenBy	George Daniel Mostow NERFINISHED ⓘ
relatedTo	Borel density theorem NERFINISHED ⓘ Thurston hyperbolization theorem NERFINISHED ⓘ quasi-isometric rigidity of lattices ⓘ
relatesConcept	fundamental group ⓘ hyperbolic metric ⓘ isometry ⓘ lattice in Lie group ⓘ locally symmetric space ⓘ quasi-isometry ⓘ
typeOfRigidity	strong rigidity GENERATED ⓘ
usedIn	classification of hyperbolic 3-manifolds ⓘ geometric group theory rigidity phenomena ⓘ study of Kleinian groups ⓘ

Referenced by (2)

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

Kleinian group → relatedTo → Mostow rigidity theorem ⓘ

Hyperbolic Manifolds and Discrete Groups → topic → Mostow rigidity theorem ⓘ

this entity surface form: Mostow rigidity