Iterated monodromy groups

Version 0.3.3

19/09/2022

Address:
Mathematisches Institut
Bunsenstraße 3-5
D-37073 Göttingen
Germany

Abstract

This document describes the package IMG, which implements in GAP the iterated monodromy groups of Nekrashevych. It depends on the package FR.

For comments or questions on IMG please contact the author; this package is still under development.

Acknowledgements

Colophon

This project started as a part of the GAP package FR. It expanded so much that I decided in 2012 to split it off, so as to keep more cleanly separated the group theory on one side, and the complex analysis, on the other side.

1 Licensing

2 IMG package

2.1 A brief mathematical introduction

2.2 An example session

3 Sphere groups and machines

3.1 Sphere groups

  3.1-1 IsSphereGroup
  3.1-2 IsomorphismSphereGroup
  3.1-3 EulerCharacteristic
  3.1-4 RankOfSphereGroup
  3.1-5 OrderingOfSphereGroup
  3.1-6 ExponentsOfSphereGroup
  3.1-7 IsomorphismFreeGroup
  3.1-8 SphereGroup
  3.1-9 IsSphereConjugacyClass
  3.1-10 IsPeripheral
  3.1-11 PeripheralClasses
  3.1-12 IntersectionNumber
  3.1-13 AutomorphismGroup
  3.1-14 AmalgamateFreeProduct

3.2 Sphere machines

  3.2-1 IsSphereMachine
  3.2-2 AsSphereMachine
  3.2-3 NewSphereMachine

3.3 Polynomial sphere machines

  3.3-1 PolynomialSphereMachine
  3.3-2 SupportingRays
  3.3-3 FRMachineWithNormalizedAdder
  3.3-4 SimplifiedSphereMachine
  3.3-5 Mating
  3.3-6 IsKneadingMachine
  3.3-7 AddingElement
  3.3-8 AsPolynomialSphereMachine
  3.3-9 LiftOfConjugacyClass
  3.3-10 ComplexConjugate
  3.3-11 RotatedSpider
  3.3-12 KneadingSequence
  3.3-13 AllInternalAddresses
  3.3-14 ExternalAnglesRelation
  3.3-15 ExternalAngle

3.4 Automorphisms of sphere machines

3.4-1 AutomorphismVirtualEndomorphism

4 Holomorphic maps

4.1 \(\mathbb P^1\) points

  4.1-1 IsP1Point
  4.1-2 P1Coordinate
  4.1-3 CleanedP1Point
  4.1-4 P1infinity
  4.1-5 P1Antipode
  4.1-6 P1Barycentre
  4.1-7 P1Circumcentre
  4.1-8 P1Distance
  4.1-9 P1Midpoint
  4.1-10 P1Sphere
  4.1-11 SphereP1
  4.1-12 SphereP1Y
  4.1-13 P1XRatio
  4.1-14 CollectedP1Points
  4.1-15 MatchP1Points
  4.1-16 ClosestP1Point
  4.1-17 IsP1Map
  4.1-18 MoebiusMap
  4.1-19 P1MapRotatingP1Points
  4.1-20 P1MapNormalizingP1Points
  4.1-21 P1z
  4.1-22 P1Monomial
  4.1-23 CleanedP1Map
  4.1-24 CompositionP1Map
  4.1-25 InverseP1Map
  4.1-26 ComplexConjugate
  4.1-27 ConjugatedP1Map
  4.1-28 CoefficientsOfP1Map
  4.1-29 P1MapByCoefficients
  4.1-30 NumeratorP1Map
  4.1-31 P1MapByZerosPoles
  4.1-32 P1Path
  4.1-33 DegreeOfP1Map
  4.1-34 P1Image
  4.1-35 P1PreImage
  4.1-36 P1PreImages
  4.1-37 Primitive
  4.1-38 P1MapScaling
  4.1-39 CriticalPointsOfP1Map
  4.1-40 AsP1Map
  4.1-41 P1MapSL2
  4.1-42 SL2P1Map
  4.1-43 SetP1Points

4.2 Triangulations

  4.2-1 IsSphereTriangulation
  4.2-2 EdgePath
  4.2-3 DelaunayTriangulation
  4.2-4 AddToTriangulation
  4.2-5 RemoveFromTriangulation
  4.2-6 LocateFaceInTriangulation
  4.2-7 WiggledTriangulation
  4.2-8 EquidistributedP1Points

4.3 Marked spheres

  4.3-1 IsMarkedSphere
  4.3-2 NewMarkedSphere
  4.3-3 Draw
  4.3-4 WiggledMarkedSphere
  4.3-5 SphereMachineOfBranchedCovering
  4.3-6 MonodromyOfP1Map
  4.3-7 SphereMachine
  4.3-8 DistanceMarkedSpheres

4.4 The Hurwitz problem

4.4-1 BranchedCoveringByMonodromy
4.4-2 DessinByPermutations

4.5 The spider algorithm

  4.5-1 NormalizedP1Map
  4.5-2 ThurstonAlgorithm
  4.5-3 SpiderAlgorithm
  4.5-4 P1MapBySphereMachine
  4.5-5 ThurstonMatrix
  4.5-6 ThurstonObstruction

5 Examples

5.1 Examples of groups

  5.1-1 PoirierExamples
  5.1-2 DBRationalIMGGroup
  5.1-3 PostCriticalMachine

6 Miscellanea

6.1 Complex numbers

6.1-1 IsPMComplex

6.2 Helpers

6.2-1 Mandel
6.2-2 NonContractingSubmatrix

6.3 User settings

6.3-1 InfoIMG

References

Index

Iterated monodromy groups

Abstract

Copyright

Acknowledgements

Colophon

Contents