SUBGROUPS OF PSL_2(R)

DETAILS

 
Introduction
      Example GrpPSL2_basic-example (H37E1)

 
Congruence Subgroups

      Creation of Subgroups of PSL_2(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
            CongruenceSubgroup(N,char) : SeqEnum, GrpDrchElt -> GrpPSL2
            Intersection(G,H) : GrpPSL2, GrpPSL2 -> GrpPSL2
            Example GrpPSL2_Creation (H37E2)

      Relations
            G eq H : GrpPSL2, GrpPSL2 -> BoolElt
            IsSubgroup(G,H) : 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 (H37E3)
      Example GrpPSL2_Element-of-congruence-subgroup-in-terms-of-generators (H37E4)

      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 (H37E5)

 
Elements of PSL_2(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 (H37E6)

 
The Upper Half Plane

      Creation
            UpperHalfPlaneWithCusps() : -> SpcHyp
            H ! x : SpcHyp, . -> SpcHypElt
            Example GrpPSL2_Upper-half-plane-example (H37E7)

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

 
Action of PSL_2(R) on the Upper Half Plane
      g * z : GrpSPL2Elt, SpcHypElt -> SpcHypElt
      z + a : SpcHypElt, RngIntElt -> 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

 
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 (H37E8)

 
Graphical Output
      DisplayPolygons(P,file) : SeqEnum, MonStgElt ->
      DisplayFareySymbolDomain(FS,file) : SymFry, MonStgElt -> SeqEnum
      Example GrpPSL2_Graphics (H37E9)
      Example GrpPSL2_more-graphics (H37E10)

 
Bibliography