Fuchsian group

E500439

Kleinian group Lie group action discrete group mathematical concept

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.

Jump to: Surface forms Statements Referenced by

Observed surface forms (2)

Surface form	Occurrences
Fuchsian groups	3
Veech group	1

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 ⓘ

Referenced by (6)

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

Kleinian group → generalizes → Fuchsian group ⓘ

Teichmüller curve → hasMonodromy → Fuchsian group ⓘ

this entity surface form: Veech group

Lazarus Fuchs → notableWork → Fuchsian group ⓘ

uniformization theorem → relatedConcept → Fuchsian group ⓘ

this entity surface form: Fuchsian groups

Hyperbolic Manifolds and Discrete Groups → topic → Fuchsian group ⓘ

this entity surface form: Fuchsian groups

Teichmüller theory → usesConcept → Fuchsian group ⓘ

this entity surface form: Fuchsian groups