modular group PSL(2,Z)

E169191

Fuchsian group arithmetic group discrete group matrix group modular group

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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (2)

Label	Occurrences
modular group PSL(2,Z) canonical	2
PSL(2,Z)	1

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 ⓘ

Referenced by (3)

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

Conway’s topograph → relatedTo → modular group PSL(2,Z) ⓘ

Farey tessellation → isInvariantUnder → modular group PSL(2,Z) ⓘ

Farey tessellation → hasSymmetryGroup → modular group PSL(2,Z) ⓘ

this entity surface form: PSL(2,Z)