Kleinian group

E259766

discrete group mathematical object subgroup of Möbius transformations

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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All labels observed (2)

Label	Occurrences
Kleinian groups	2
Kleinian group canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T2364474 — 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: Kleinian group
Context triple: [Riemann surface, relatedTo, Kleinian group]

A. 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.
B. 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.
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. SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
E. Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
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: Kleinian group
Target entity description: 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.

A. 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.
B. 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.
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. SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
E. Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	discrete group ⓘ mathematical object ⓘ subgroup of Möbius transformations ⓘ
actsBy	Möbius transformations ⓘ
actsOn	Riemann sphere ⓘ hyperbolic 3-space ⓘ
associatedWith	Riemann surfaces ⓘ surface form: Riemann surface hyperbolic 3-manifold ⓘ
classificationBy	conformal boundary ⓘ geometric finiteness ⓘ limit set type ⓘ
context	PSL(2,ℂ)-geometry ⓘ discrete subgroups of Lie groups ⓘ
field	complex analysis ⓘ geometric group theory ⓘ hyperbolic geometry ⓘ low-dimensional topology ⓘ
generalizes	Fuchsian group ⓘ
hasDimensionOfAction	3 ⓘ
hasDomainOfDiscontinuity	open subset of Riemann sphere ⓘ
hasInvariant	conformal dimension of limit set ⓘ critical exponent of Poincaré series ⓘ limit set Hausdorff dimension ⓘ volume of associated hyperbolic 3-manifold ⓘ
hasLimitSet	subset of Riemann sphere ⓘ
hasProperty	can be convex cocompact ⓘ can be elementary ⓘ can be finitely generated ⓘ can be geometrically finite ⓘ can be nonelementary ⓘ properly discontinuous action on domain of discontinuity ⓘ
hasRepresentation	discrete faithful representation into PSL(2,ℂ) ⓘ
is	discrete subgroup of PSL(2,ℂ) ⓘ
namedAfter	Felix Klein ⓘ
parameterSpace	character variety ⓘ deformation space of Kleinian groups ⓘ
relatedTo	Ahlfors finiteness theorem ⓘ Mostow rigidity theorem ⓘ Sullivan dictionary ⓘ Thurston hyperbolization theorem ⓘ deformation theory of hyperbolic structures ⓘ quasiconformal mappings ⓘ
specialCase	Fuchsian group acting on hyperbolic 2-space ⓘ
studiedIn	3-manifold theory ⓘ Teichmüller theory ⓘ complex dynamics ⓘ
studiedUsing	conformal geometry ⓘ dynamical systems ⓘ ergodic theory ⓘ

How these facts were elicited

Referenced by (3)

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

Riemann surfaces → relatedTo → Kleinian group ⓘ

subject surface form: Riemann surface

uniformization theorem → relatedConcept → Kleinian group ⓘ

this entity surface form: Kleinian groups

Hyperbolic Manifolds and Discrete Groups → topic → Kleinian group ⓘ