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
Introduction
Example GrpPSL2_basic-example (H142E1)
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 (H142E2)
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 (H142E3)
Example GrpPSL2_Element-of-congruence-subgroup-in-terms-of-generators (H142E4)
Cusps and Elliptic Points of Congruence Subgroups
Cusps(G) : GrpPSL2 -> SeqEnum
CuspWidth(G,x) : GrpPSL2, SetCspElt -> RngIntElt
EllipticPoints(G) : GrpPSL2 -> [SpcHypElt]
Example GrpPSL2_cusp-example (H142E5)
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 (H142E6)
The Upper Half Plane
Creation
UpperHalfPlane() : -> SpcHyp
H ! x : SpcHyp, . -> SpcHypElt
Example GrpPSL2_Upper-half-plane-example (H142E7)
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 (H142E8)
Graphical Output
DisplayPolygons(P,file) : SeqEnum, MonStgElt ->
DisplayFareySymbolDomain(FS,file) : SymFry, MonStgElt -> SeqEnum
Example GrpPSL2_Graphics (H142E9)
Example GrpPSL2_more-graphics (H142E10)
Example GrpPSL2_Graphics (H142E11)
Bibliography
[Next][Prev] [Right] [____] [Up] [Index] [Root]
|