PSL(2,ℝ)
E656693
Lie group
center-free group
connected Lie group
linear group
rank-one Lie group
real Lie group
real algebraic group
semisimple Lie group
simple Lie group
PSL(2,ℝ) is the Lie group of orientation-preserving isometries of the hyperbolic plane, realized as 2×2 real matrices with determinant 1 modulo their center.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
center-free group ⓘ connected Lie group ⓘ linear group ⓘ rank-one Lie group ⓘ real Lie group ⓘ real algebraic group ⓘ semisimple Lie group ⓘ simple Lie group ⓘ |
| actionType | orientation-preserving isometries of hyperbolic plane ⓘ |
| actsOn |
Poincaré disk model of hyperbolic plane
NERFINISHED
ⓘ
upper half-plane model of hyperbolic plane ⓘ |
| CartanDecomposition | KAK with K≅SO(2) ⓘ |
| definedAs | SL(2,ℝ)/{±I} NERFINISHED ⓘ |
| dimensionOverℝ | 3 ⓘ |
| fullName | projective special linear group of 2×2 real matrices NERFINISHED ⓘ |
| fundamentalGroup | ℤ ⓘ |
| hasCenter | trivial group ⓘ |
| hasKillingFormSignature | (2,1) ⓘ |
| hasLatticeSubgroups | fundamental groups of closed hyperbolic surfaces ⓘ |
| hasLieAlgebra | sl(2,ℝ) NERFINISHED ⓘ |
| hasMaximalCompactSubgroup | SO(2) NERFINISHED ⓘ |
| hasPropertyT | false ⓘ |
| hasRealFormOf | complex Lie group PSL(2,ℂ) ⓘ |
| isAmenable | false ⓘ |
| isConnected | true ⓘ |
| isDoubleCoverOf | SO⁺(2,1) NERFINISHED ⓘ |
| isFuchsianGroupContainer | contains discrete subgroups called Fuchsian groups ⓘ |
| isGromovHyperbolic | false ⓘ |
| isIsometryGroupOf | hyperbolic plane ⓘ |
| isNonCompact | true ⓘ |
| isomorphicTo |
Isom⁺(ℍ²)
NERFINISHED
ⓘ
orientation-preserving isometry group of hyperbolic plane ⓘ |
| isRealPointsOf | algebraic group PSL₂ over ℝ NERFINISHED ⓘ |
| isSimplyConnected | false ⓘ |
| isWordHyperbolic | false ⓘ |
| IwasawaDecomposition | KAN with K≅SO(2), A≅ℝ, N≅ℝ ⓘ |
| locallyIsomorphicTo |
SL(2,ℝ)
NERFINISHED
ⓘ
SO⁺(2,1) NERFINISHED ⓘ |
| quotientBy | {±I} ⓘ |
| quotientOf | SL(2,ℝ) NERFINISHED ⓘ |
| realRank | 1 ⓘ |
| universalCover | universal covering group of SL(2,ℝ) ⓘ |
| usedIn |
Teichmüller theory
NERFINISHED
ⓘ
automorphic forms ⓘ hyperbolic geometry ⓘ representation theory of Lie groups ⓘ theory of Fuchsian groups ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
PSL(2,ℤ)
subject surface form:
PSL(2,ℤ)