Fuchsian group
E500439
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Fuchsian groups | 3 |
| Fuchsian group canonical | 2 |
| Veech group | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5176470 — 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: Fuchsian group Context triple: [Lazarus Fuchs, notableWork, Fuchsian group]
-
A.
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.
-
B.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
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C.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
D.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
E.
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.
- 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: Fuchsian group Target entity description: A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
-
A.
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.
-
B.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
-
C.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
D.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
E.
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.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
Kleinian group
ⓘ
Lie group action ⓘ discrete group ⓘ mathematical concept ⓘ |
| actsBy | isometries ⓘ |
| actsOn | hyperbolic plane ⓘ |
| correspondsTo |
hyperbolic Riemann surface
ⓘ
hyperbolic orbifold ⓘ quotient of hyperbolic plane by group action ⓘ |
| hasInvariant |
Hausdorff dimension of limit set
ⓘ
covolume in PSL(2,R) ⓘ limit set on the boundary of the hyperbolic plane ⓘ |
| hasModel |
Poincaré disk model of hyperbolic plane
NERFINISHED
ⓘ
upper half-plane model of hyperbolic plane ⓘ |
| hasProperty |
can be co-compact
ⓘ
can be finitely generated ⓘ can be infinitely generated ⓘ can be non-co-compact ⓘ can be of the first kind ⓘ can be of the second kind ⓘ can be torsion-free ⓘ can contain elliptic elements ⓘ can contain hyperbolic elements ⓘ can contain parabolic elements ⓘ can have finite covolume ⓘ properly discontinuous action on the hyperbolic plane ⓘ |
| hasSubClass |
arithmetic Fuchsian group
ⓘ
lattice in PSL(2,R) ⓘ surface group representation into PSL(2,R) ⓘ triangle group ⓘ |
| is |
discrete subgroup of PSL(2,R)
ⓘ
discrete subgroup of orientation-preserving isometries of the hyperbolic plane ⓘ |
| isAnalogOf | Kleinian group acting on hyperbolic 3-space ⓘ |
| isDefinedAs |
discrete group of orientation-preserving isometries of the hyperbolic plane
ⓘ
discrete subgroup of PSL(2,R) acting by Möbius transformations on the upper half-plane ⓘ |
| isExampleOf | discrete subgroup of a Lie group ⓘ |
| isGeneralizationOf | modular group ⓘ |
| isRelatedTo |
fundamental group of a Riemann surface
ⓘ
modular group PSL(2,Z) NERFINISHED ⓘ |
| isUsedIn |
Teichmüller theory
ⓘ
automorphic forms ⓘ complex analysis ⓘ differential geometry ⓘ geometric group theory ⓘ hyperbolic geometry ⓘ low-dimensional topology ⓘ modular forms ⓘ theory of Riemann surfaces ⓘ |
| namedAfter | Lazarus Fuchs NERFINISHED ⓘ |
| studiedIn |
Fuchsian groups and automorphic functions
ⓘ
Teichmüller space theory ⓘ |
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: Fuchsian group Description of subject: A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fuchsian groups
this entity surface form:
Fuchsian groups
this entity surface form:
Veech group
this entity surface form:
Fuchsian groups