Mostow rigidity theorem
E898487
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Mostow rigidity | 2 |
| Mostow rigidity theorem canonical | 1 |
| Mostow strong rigidity theorem | 1 |
| Mostow–Prasad rigidity theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991687 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Mostow rigidity theorem Context triple: [Kleinian group, relatedTo, Mostow rigidity theorem]
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A.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
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B.
Calabi conjecture
The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
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C.
Cheeger–Gromov compactness theorem
The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
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D.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
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E.
Conformal Invariants
Conformal Invariants is a foundational mathematical work by Lars Ahlfors that systematically develops the theory of quantities preserved under conformal mappings in complex analysis and geometric function theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Mostow rigidity theorem Target entity description: 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.
-
A.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
-
B.
Calabi conjecture
The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
-
C.
Cheeger–Gromov compactness theorem
The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
-
D.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
-
E.
Conformal Invariants
Conformal Invariants is a foundational mathematical work by Lars Ahlfors that systematically develops the theory of quantities preserved under conformal mappings in complex analysis and geometric function theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in differential geometry ⓘ result in geometric topology ⓘ rigidity theorem ⓘ |
| alsoKnownAs |
Mostow strong rigidity theorem
ⓘ
linked to:
Mostow rigidity theorem
|
| 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
ⓘ
linked to:
Riemannian manifolds
geometric group theory ⓘ geometry ⓘ hyperbolic geometry ⓘ topology ⓘ |
| generalizationOf | rigidity results for lattices in higher-rank Lie groups ⓘ |
| hasGeneralization |
Margulis superrigidity theorem
ⓘ
Prasad rigidity results ⓘ |
| 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
ⓘ
linked to:
George D. Mostow
|
| provenBy |
George Daniel Mostow
ⓘ
linked to:
George D. Mostow
|
| relatedTo |
Borel density theorem
ⓘ
Thurston hyperbolization theorem ⓘ quasi-isometric rigidity of lattices ⓘ |
| relatesConcept |
fundamental group
ⓘ
hyperbolic metric ⓘ isometry ⓘ lattice in Lie group ⓘ locally symmetric space ⓘ quasi-isometry ⓘ |
| typeOfRigidity | strong rigidity ⓘ |
| usedIn |
classification of hyperbolic 3-manifolds
ⓘ
geometric group theory rigidity phenomena ⓘ study of Kleinian groups ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Mostow rigidity theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Mostow rigidity theorem
linked to: Mostow rigidity theorem
linked to: Mostow rigidity theorem
linked to: Mostow rigidity theorem