Introduction
Example GrpPSL2_basic-example (H130E1)
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)
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)
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
Wed Apr 24 15:09:57 EST 2013