modular group PSL(2,Z)
E169191
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| modular group PSL(2,Z) canonical | 2 |
| PSL(2,Z) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1484054 — 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: modular group PSL(2,Z) Context triple: [Conway’s topograph, relatedTo, modular group PSL(2,Z)]
-
A.
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.
-
B.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
D.
Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
-
E.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
- 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: modular group PSL(2,Z) Target entity description: 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.
-
A.
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.
-
B.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
D.
Conway groups
Conway groups are a set of three closely related sporadic simple groups discovered by John H. Conway in the study of symmetries of the Leech lattice in group theory.
-
E.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Fuchsian group
ⓘ
arithmetic group ⓘ discrete group ⓘ matrix group ⓘ modular group ⓘ |
| actionFormula | z ↦ (az + b)/(cz + d) for matrix [[a,b],[c,d]] ⓘ |
| actsBy | fractional linear transformations ⓘ |
| actsOn | upper half-plane ℍ ⓘ |
| actsProperlyDiscontinuouslyOn | upper half-plane ℍ ⓘ |
| cofiniteVolumeIn | PSL(2,ℝ) ⓘ |
| commensurableWith | SL(2,ℤ) ⓘ |
| containsSubgroup |
principal congruence subgroup Γ(N)
ⓘ
Γ(2) ⓘ Γ₀(N) ⓘ Γ₁(N) ⓘ |
| definedAs | SL(2,ℤ)/{±I} ⓘ |
| fundamentalDomain | {z ∈ ℍ : |Re(z)| ≤ 1/2, |z| ≥ 1} ⓘ |
| generatedBy |
S
ⓘ
T ⓘ |
| generatorAction |
S:z ↦ -1/z
ⓘ
T:z ↦ z+1 ⓘ |
| hasAbelianization | C₆ ⓘ |
| hasCenter | trivial group ⓘ |
| hasCocompactLatticeProperty | false ⓘ |
| hasCusp | ∞ ⓘ |
| hasElementOfOrder |
2
ⓘ
3 ⓘ ∞ ⓘ |
| hasFiniteAreaQuotient | ℍ/PSL(2,ℤ) ⓘ |
| hasQuotient | PSL(2,ℤ/Nℤ) ⓘ |
| hasTorsion | true ⓘ |
| hasUnderlyingSet | 2×2 integer matrices with determinant 1 modulo ±I ⓘ |
| hasWordProblem | decidable ⓘ |
| isCountable | true ⓘ |
| isFinitelyGenerated | true ⓘ |
| isFinitelyPresented | true ⓘ |
| isLatticeIn | PSL(2,ℝ) ⓘ |
| isNonAbelian | true ⓘ |
| isomorphicTo | free product C₂ * C₃ ⓘ |
| isQuotientOf | SL(2,ℤ) ⓘ |
| kernelOfProjectionFrom | {±I} in SL(2,ℤ) ⓘ |
| presentation | ⟨S,T | S² = 1, (ST)³ = 1⟩ ⓘ |
| quotientIs | modular orbifold ⓘ |
| rankOverℤ | 2 as free product C₂ * C₃ ⓘ |
| relatedTo |
Riemann surfaces
ⓘ
Teichmüller theory ⓘ automorphic forms ⓘ elliptic curves ⓘ modular forms ⓘ |
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: modular group PSL(2,Z) Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: modular group PSL(2,Z)