SL2(R)
SL(2, R) acts on the complex upper half-plane by fractional linear transformations.By the Riemann mapping theorem, it is also isomorphic to the group of conformal automorphisms of the unit disc.These Möbius transformations act as the isometries of the upper half-plane model of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the Poincaré disk model.The above formula can be also used to define Möbius transformations of dual and double (aka split-complex) numbers.The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R).The result is the following representation: The Killing form on sl(2, R) has signature (2,1), and induces an isomorphism between PSL(2, R) and the Lorentz group SO+(2,1).The eigenvalues of an element A ∈ SL(2, R) satisfy the characteristic polynomial and therefore This leads to the following classification of elements, with corresponding action on the Euclidean plane: The names correspond to the classification of conic sections by eccentricity: if one defines eccentricity as half the absolute value of the trace (ε = 1/2 |tr|; dividing by 2 corrects for the effect of dimension, while absolute value corresponds to ignoring an overall factor of ±1 such as when working in PSL(2, R)), then this yields:However, each element is conjugate to a member of one of 3 standard one-parameter subgroups (possibly times ±1), as detailed below.The eigenvalues for an elliptic element are both complex, and are conjugate values on the unit circle.These are all the elements of the modular group with finite order, and they act on the torus as periodic diffeomorphisms.Such an element acts as a shear mapping on the Euclidean plane, and the corresponding element of PSL(2, R) acts as a limit rotation of the hyperbolic plane and as a null rotation of Minkowski space.Parabolic elements of the modular group act as Dehn twists of the torus.Parabolic elements are conjugate into the 2 component group of standard shears × ±I:Such an element acts as a squeeze mapping of the Euclidean plane, and the corresponding element of PSL(2, R) acts as a translation of the hyperbolic plane and as a Lorentz boost on Minkowski space.Hyperbolic elements of the modular group act as Anosov diffeomorphisms of the torus.Hyperbolic elements are conjugate into the 2 component group of standard squeezes × ±I:As a topological space, PSL(2, R) can be described as the unit tangent bundle of the hyperbolic plane.SL(2, R) is a 2-fold cover of PSL(2, R), and can be thought of as the bundle of spinors on the hyperbolic plane.is the universal cover of the unit tangent bundle to any hyperbolic surface.is orientable, and is a circle bundle over some 2-dimensional hyperbolic orbifold (a Seifert fiber space).The center of SL(2, R) is the two-element group {±1}, and the quotient PSL(2, R) is simple.The most famous of these is the modular group PSL(2, Z), which acts on a tessellation of the hyperbolic plane by ideal triangles.The group has several families of unitary representations, which were worked out in detail by Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
Algebraic structureGroup theorySubgroupNormal subgroupGroup actionQuotient group(Semi-)direct productDirect sumFree productWreath productGroup homomorphismssimplefiniteinfinitecontinuousmultiplicativeadditivecyclicabeliandihedralnilpotentsolvableGlossary of group theoryList of group theory topicsFinite groupsCyclic groupSymmetric groupAlternating groupDihedral groupQuaternion groupCauchy's theoremLagrange's theoremSylow theoremsHall's theoremp-groupElementary abelian groupFrobenius groupSchur multiplierClassification of finite simple groupsLie typesporadicDiscrete groupsLatticesIntegersFree groupModular groupsArithmetic groupLatticeHyperbolic groupTopologicalLie groupsSolenoidCircleGeneral linearSpecial linearOrthogonalEuclideanSpecial orthogonalUnitarySpecial unitarySymplecticLorentzPoincaréConformalDiffeomorphismInfinite dimensional Lie groupAlgebraic groupsLinear algebraic groupReductive groupAbelian varietyElliptic curvemathematicsspecial linear groupmatricesdeterminantconnectednon-compactLie groupgeometrytopologyrepresentation theoryphysicscomplex upper half-planefractional linear transformationsquotientprojective special linear groupidentity matrixmodular groupcovering groupmetaplectic groupsymplectic grouplinear transformationsorientedisomorphiccoquaternionsorientationprojective transformationsreal projective lineautomorphismsunit discisometrieshyperbolic planeLorentz groupMinkowski spaceindefinite orthogonal groupspin grouphomographiesRiemann sphereMöbius transformationshyperbolic motionsupper half-planeRiemann mapping theoremupper half-plane modelPoincaré disk modeldouble (aka split-complex) numbersLobachevskian geometryconjugationrepresentationquadratic formsKilling formsignaturehyperboloid modeleigenvaluescharacteristic polynomialrotation