Hyperbolic Manifolds and Discrete Groups
E325284
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hyperbolic Manifolds and Discrete Groups canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3072662 — 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: Hyperbolic Manifolds and Discrete Groups Context triple: [Annals of Mathematics Studies, hasNotableWork, Hyperbolic Manifolds and Discrete Groups]
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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.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
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C.
Dehn’s decision problems in group theory
Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
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D.
Dehn complex
The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
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E.
Teichmüller theory
Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
- 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: Hyperbolic Manifolds and Discrete Groups Target entity description: "Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
-
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.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
-
C.
Dehn’s decision problems in group theory
Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
-
D.
Dehn complex
The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
-
E.
Teichmüller theory
Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematics book
ⓘ
monograph ⓘ nonfiction book ⓘ |
| audience |
graduate students in mathematics
ⓘ
research mathematicians ⓘ |
| characteristic |
focus on low-dimensional phenomena
ⓘ
foundational treatment of hyperbolic manifolds ⓘ systematic development of discrete group actions ⓘ |
| emphasis |
applications to low-dimensional topology
ⓘ
connections between hyperbolic geometry and discrete groups ⓘ |
| field |
Kleinian groups
ⓘ
Riemannian geometry ⓘ discrete groups ⓘ geometric group theory ⓘ geometric topology ⓘ hyperbolic geometry ⓘ low-dimensional topology ⓘ |
| topic |
3-manifolds
ⓘ
Fuchsian group ⓘ
surface form:
Fuchsian groups
Kleinian group ⓘ
surface form:
Kleinian groups
Mostow rigidity theorem ⓘ
surface form:
Mostow rigidity
Lambert series ⓘ
surface form:
Poincaré series
Teichmüller theory ⓘ cofinite volume groups ⓘ convex cores of hyperbolic manifolds ⓘ covering spaces ⓘ cusps of hyperbolic manifolds ⓘ deformation spaces of discrete groups ⓘ discrete group actions ⓘ ends of hyperbolic manifolds ⓘ ergodic theory of group actions ⓘ fundamental groups of manifolds ⓘ geodesics in hyperbolic manifolds ⓘ geometric structures on manifolds ⓘ growth of groups ⓘ hyperbolic manifolds ⓘ isometries of hyperbolic space ⓘ lattices in Lie groups ⓘ limit sets of discrete groups ⓘ low-dimensional manifolds ⓘ orbifolds ⓘ quotients of hyperbolic space ⓘ spectral theory on hyperbolic manifolds ⓘ volume of hyperbolic manifolds ⓘ |
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: Hyperbolic Manifolds and Discrete Groups Description of subject: "Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.