Congruence Subgroups of \({\operatorname{PSL}}_2({\mathbb{R}})\)

• Introduction
  • Example: Basic Example
• Congruence Subgroups
  • Creation of Subgroups of \({\operatorname{PSL}}_2({\mathbb{R}})\)
    • 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
    • G meet H: GrpPSL2, GrpPSL2 → GrpPSL2
    • Example: Creation
  • 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 Of Finding Coset Representatives
  • Example: Element Of Congruence Subgroup In Terms Of Generators
  • Cusps and Elliptic Points of Congruence Subgroups
    • Cusps(G): GrpPSL2 → SeqEnum
    • CuspWidth(G,x): GrpPSL2, SetCspElt → RngIntElt
    • EllipticPoints(G): GrpPSL2 → [SpcHypElt]
    • EllipticPoints(G,H): GrpPSL2, SpcHyp → [SpcHypElt]
    • Example: Cusp Example
• Elements of \({\operatorname{PSL}}_2({\mathbb{R}})\)
  • Creation
    • G ! x: GrpPSL2, . → GrpPSL2
    • G ! x: GrpPSL2, RngIntElt → GrpPSL2
    • G ! x: GrpPSL2, GrpMatElt → GrpPSL2
    • G ! x: GrpPSL2, AlgMatElt → GrpPSL2
    • G ! x: GrpPSL2, GrpPSL2Elt → 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: Creation CongruenceSubgroups
• The Upper Half Plane
  • Creation
    • UpperHalfPlane() → SpcHyp
    • H ! x: SpcHyp, . → SpcHypElt
    • Example: Upper Half Plane Example
  • 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 \({\operatorname{PSL}}_2({\mathbb{R}})\) on the Upper Half Plane
  • g * z: GrpPSL2Elt, SpcHypElt → SpcHypElt
  • g * z: GrpPSL2Elt, [SpcHypElt] → [SpcHypElt]
  • g * z: GrpPSL2Elt, SetCspElt → SetCspElt
  • g * z: RngIntElt, SpcHypElt → SpcHypElt
  • g * z: RngIntElt, [SpcHypElt] → [SpcHypElt]
  • g * z: RngIntElt, SetCspElt → SetCspElt
  • 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
    • z + a: SpcHypElt, FldRatElt → SpcHypElt
    • z - a: SpcHypElt, RngIntElt → SpcHypElt
    • z - a: SpcHypElt, FldRatElt → SpcHypElt
    • a * z: RngElt, SpcHypElt → SpcHypElt
    • z * a: SpcHypElt, RngElt → SpcHypElt
    • a * z: RngIntElt, SetCspElt → SetCspElt
    • a * z: FldRatElt, SetCspElt → SetCspElt
    • a * seq: RngElt, [SpcHypElt] → [SpcHypElt]
    • a * z: RngIntElt, [SetCspElt] → [SetCspElt]
    • a * z: FldRatElt, [SetCspElt] → [SetCspElt]
    • z * a: SpcHypElt, RngIntElt → SpcHypElt
    • z * a: SpcHypElt, RngIntElt → SpcHypElt
    • z * a: SpcHypElt, RngIntElt → SpcHypElt
    • z * a: SpcHypElt, RngIntElt → SpcHypElt
    • z * a: SpcHypElt, RngIntElt → SpcHypElt
    • z / a: SpcHypElt, RngIntElt → 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
    • GeodesicsIntersection(x1,x2): [SetCspElt], [SetCspElt]) → 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
  • FundamentalDomain(FS,H): SymFry, SpcHyp → SeqEnum
  • CosetRepresentatives(FS): SymFry → SeqEnum
  • InternalEdges(FS): SymFry → SeqEnum
• Points and Geodesics
  • GeodesicsIntersection(x,y): [SpcHypElt], [SpcHypElt] → SpcHypElt
  • GeodesicsIntersection(x,y): [SetCspElt], [SetCspElt] → SpcHypElt
  • Example: Geodesic Intersection
• Graphical Output
  • DisplayPolygons(P,file): SeqEnum, MonStgElt
  • DisplayFareySymbolDomain(FS,file): SymFry, MonStgElt → SeqEnum
  • DisplayFareySymbolDomain(G,file): GrpPSL2, MonStgElt → SeqEnum
  • Example: Graphics
  • Example: More Graphics
  • Example: Graphics