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CONGRUENCE SUBGROUPS OF PSL2(R)

 
Acknowledgements
 
Introduction
 
Congruence Subgroups
      Creation of Subgroups of PSL2(R)
      Relations
      Basic Attributes
 
Structure of Congruence Subgroups
      Cusps and Elliptic Points of Congruence Subgroups
 
Elements of PSL2(R)
      Creation
      Membership and Equality Testing
      Basic Functions
 
The Upper Half Plane
      Creation
      Basic Attributes
 
Action of PSL2(R) on the Upper Half Plane
      Arithmetic
      Distances, Angles and Geodesics
 
Farey Symbols and Fundamental Domains
 
Points and Geodesics
 
Graphical Output
 
Bibliography







DETAILS

 
Introduction
      Example GrpPSL2_basic-example (H130E1)

 
Congruence Subgroups

      Creation of Subgroups of PSL2(R)
            PSL2(R) : Rng -> GrpPSL2
            Gamma0(N) : RngIntElt -> GrpPSL2
            Gamma1(N) : RngIntElt -> GrpPSL2
            GammaUpper0(N) : RngIntElt -> GrpPSL2
            GammaUpper1(N) : RngIntElt -> GrpPSL2
            CongruenceSubgroup(N) : RngIntElt -> GrpPSL2
            CongruenceSubgroup(i,N) : RngIntElt, RngIntElt -> GrpPSL2
            CongruenceSubgroup([N,M,P]) : SeqEnum -> GrpPSL2
            Intersection(G,H) : GrpPSL2, GrpPSL2 -> GrpPSL2
            Example GrpPSL2_Creation (H130E2)

      Relations
            G eq H : GrpPSL2, GrpPSL2 -> BoolElt
            H subset G : GrpPSL2, GrpPSL2 -> BoolElt
            Index(G,H) : GrpPSL2, GrpPSL2 -> RngIntElt
            Index(G) : GrpPSL2 -> RngIntElt

      Basic Attributes
            Level(G) : GrpPSL2 -> RngIntElt
            IsCongruence(G) : GrpPSL2 -> BoolElt
            IsGamma0(G) : GrpPSL2 -> BoolElt
            IsGamma1(G) : GrpPSL2 -> BoolElt
            BaseRing(G) : GrpPSL2 -> Rng
            Identity(G) : GrpPSL2 -> GrpPSL2Elt

 
Structure of Congruence Subgroups
      CosetRepresentatives(G) : GrpPSL2 -> SeqEnum
      Generators(G) : GrpPSL2 -> SeqEnum
      FindWord(G,g) : GrpPSL2, GrpPSL2Elt -> SeqEnum
      Genus(G) : GrpPSL2 -> RngIntElt
      FundamentalDomain(G) : GrpPSL2 -> SeqEnum
      Example GrpPSL2_Example-of-finding-coset-representatives (H130E3)
      Example GrpPSL2_Element-of-congruence-subgroup-in-terms-of-generators (H130E4)

      Cusps and Elliptic Points of Congruence Subgroups
            Cusps(G) : GrpPSL2 -> SeqEnum
            CuspWidth(G,x) : GrpPSL2, SetCspElt -> RngIntElt
            EllipticPoints(G) : GrpPSL2, SpcHyp -> [SpcHypElt]
            Example GrpPSL2_cusp-example (H130E5)

 
Elements of PSL2(R)

      Creation
            G ! x : GrpPSL2, . -> GrpPSL2
            Random(G,m) : GrpPSL2, RngIntElt -> GrpPSL2Elt

      Membership and Equality Testing
            g eq h : GrpPSL2Elt, GrpPSL2Elt -> BoolElt
            IsEquivalent(g,h,G) : GrpPSL2Elt, GrpPSL2Elt, GrpPSL2 -> BoolElt
            g in G : GrpPSL2Elt, GrpPSL2 -> BoolElt

      Basic Functions
            Eltseq(g) : GrpPSL2Elt -> SeqEnum
            g * h : GrpPSL2Elt, GrpPSL2Elt -> GrpPSL2Elt
            g ^ n : GrpPSL2Elt, RngIntElt -> GrpPSL2Elt
            Example GrpPSL2_Creation-CongruenceSubgroups (H130E6)

 
The Upper Half Plane

      Creation
            UpperHalfPlane() : -> SpcHyp
            H ! x : SpcHyp, . -> SpcHypElt
            Example GrpPSL2_Upper-half-plane-example (H130E7)

      Basic Attributes
            Imaginary(z) : SpcHypElt -> FldReElt
            Real(z) : SpcHypElt -> FldReElt
            IsReal(z) : SpcHypElt -> BoolElt
            IsCusp(z) : SpcHypElt -> BoolElt
            IsInfinite(z) : SpcHypElt -> BoolElt
            IsExact(z) : SpcHypElt -> BoolElt
            ExactValue(z) : SpcHypElt -> .
            ComplexValue(x) : SpcHypElt -> FldComElt
            x eq y : SpcHypElt, SpcHypElt -> BoolElt

 
Action of PSL2(R) on the Upper Half Plane
      g * z : GrpPSL2Elt, SpcHypElt -> SpcHypElt
      FixedPoints(g,H) : GrpPSL2Elt, SpcHyp -> SeqEnum
      IsEquivalent(G,a,b) : GrpPSL2, SpcHypElt, SpcHypElt -> BoolElt, GrpPSL2Elt
      EquivalentPoint(x) : SpcHypElt -> SpcHypElt, GrpPSL2Elt
      Stabilizer(a,G) : SpcHypElt, GrpPSL2 -> GrpPSL2Elt
      FixedArc(g,H) : GrpPSL2Elt, SpcHyp -> SeqEnum

      Arithmetic
            z + a : SpcHypElt, RngIntElt -> SpcHypElt
            a * z : RngElt, SpcHypElt -> SpcHypElt

      Distances, Angles and Geodesics
            Distance(z,w) : SpcHypElt, SpcHypElt -> FldReElt
            TangentAngle(x,y) : SpcHypElt, SpcHypElt -> FldReElt
            Angle(e1,e2) : [SpcHypElt], [SpcHypElt] -> FldReElt
            ExtendGeodesic([z1,z2], H) : [SpcHypElt], SpcHyp -> [SpcHypElt]
            GeodesicsIntersection(x1,x2) : [SpcHypElt], [SpcHypElt]) -> SeqEnum

 
Farey Symbols and Fundamental Domains
      FareySymbol(G) : GrpPSL2 -> SymFry
      Cusps(FS) : SymFry -> SeqEnum
      Labels(FS) : SymFry-> SeqEnum
      Generators(FS) : SymFry -> SeqEnum
      Group(FS) : SymFry -> GrpPSL2
      Widths(FS) : SymFry -> SeqEnum
      Index(FS) : SymFry -> RngIntElt
      FundamentalDomain(FS) : SymFry -> SeqEnum
      CosetRepresentatives(FS) : SymFry -> SeqEnum
      InternalEdges(FS) : SymFry -> SeqEnum

 
Points and Geodesics
      GeodesicsIntersection(x,y) : [SpcHypElt],[SpcHypElt] -> SpcHypElt
      Example GrpPSL2_geodesic-intersection (H130E8)

 
Graphical Output
      DisplayPolygons(P,file) : SeqEnum, MonStgElt ->
      DisplayFareySymbolDomain(FS,file) : SymFry, MonStgElt -> SeqEnum
      Example GrpPSL2_Graphics (H130E9)
      Example GrpPSL2_more-graphics (H130E10)
      Example GrpPSL2_Graphics (H130E11)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012