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# Functionally recursive groups

## Self-similar groups

Version 0.857142p7

$Date: 2008/11/13 15:28:12 $

Groups generated by automata or satisfying functional recursions

Laurent Bartholdi
Email: laurent dot bartholdi at gmail dot com
Homepage: http://www.uni-math.gwdg.de/laurent/

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

### Abstract

This document describes the package FR, which implements in GAP the basic objects of Mealy machines and functional recursions; and handles groups that they generate.

The computer algebra system GAP is available at http://gap-system.org.

This documentation for FR is available at http://www.uni-math.gwdg.de/laurent/FR/manual.pdf in PDF format, and may be accessed online at http://www.uni-math.gwdg.de/laurent/FR/.

The latest source of the package may be downloaded as http://www.uni-math.gwdg.de/laurent/FR/fr.tar.gz (tar, gzipped) and http://www.uni-math.gwdg.de/laurent/FR/fr.zoo (zoo).

Groups defined by a recursive action on a rooted tree can be defined in GAP via their recursion. Various algorithms are implemented to manipulate these groups and their elements.

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

### Copyright

© 2006-2008 by Laurent Bartholdi

### Acknowledgements

Part of this work is supported by the "Swiss National Fund for Scientific Research"

### Colophon

This project started in the mid-1990s, when, as a PhD student I did many calculations with groups generated by automata, and realized the similarities between all calculations; it quickly became clear that these calculations could be done much better by a computer than by a human.

The first routines I wrote constructed finite representations of the groups considered, so as to get insight from fast calculations within GAP. The results then had to be proved correct within the infinite group under consideration, and this often involved guessing appropriate words in the infinite group with a given image in the finite quotient.

Around 2000, I had developed quite a few routines, which I assembled in a GAP package, that dealt directly with infinite groups. This package was primitive at its core, but was extended with various routines as they became useful.

I decided in late 2005 to start a new package from scratch, that would encorporate as much functionality as possible in a uniform manner; that would handle semigroups as well as groups; that could be easily extended; and with a complete, understandable documentation. I hope I am not too far from these objectives.

### Contents

1 Licensing
2 FR package
2.1 A brief mathematical introduction
2.2 An example session
3 Functionally recursive machines
3.1 Types of machines
3.2 Products of machines
3.3 Creators for `FRMachine`s
3.3-1 FRMachineNC
3.3-2 FRMachine
3.3-3 UnderlyingFRMachine
3.3-4 AsGroupFRMachine
3.3-5 ChangeFRMachineBasis
3.4 Attributes for `FRMachine`s
3.4-1 StateSet
3.4-2 GeneratorsOfFRMachine
3.4-3 Output
3.4-4 Transition
3.4-5 WreathRecursion
3.5 Operations for `FRMachine`s
3.5-1 StructuralGroup
3.5-2 \+
3.5-3 \*
3.5-4 TensorSumOp
3.5-5 TensorProductOp
3.5-6 DirectSumOp
3.5-7 DirectProductOp
3.5-8 TreeWreathProduct
3.5-9 SubFRMachine
3.5-10 Minimized
3.5-11 Correspondence
4 Functionally recursive elements
4.1 Creators for `FRElement`s
4.1-1 FRElementNC
4.1-2 FRElement
4.1-3 FRElement
4.1-4 ComposeElement
4.1-5 VertexElement
4.1-6 DiagonalElement
4.1-7 AsGroupFRElement
4.2 Operations and Attributes for `FRElement`s
4.2-1 Output
4.2-2 Activity
4.2-3 Transition
4.2-4 Portrait
4.2-5 DecompositionOfFRElement
4.2-6 StateSet
4.2-7 State
4.2-8 States
4.2-9 FixedStates
4.2-10 LimitStates
4.2-11 IsFiniteStateFRElement
4.2-12 InitialState
4.2-13 \^
4.2-14 \*
4.2-15 \[\]
5 Mealy machines and elements
5.1 Creators for `MealyMachine`s and `MealyElement`s
5.1-1 MealyMachine
5.1-2 MealyMachine
5.1-3 MealyMachineNC
5.1-4 AllMealyMachines
5.2 Operations and Attributes for `MealyMachine`s and `MealyElement`s
5.2-1 Draw
5.2-2 Minimized
5.2-3 DualMachine
5.2-4 IsReversible
5.2-5 IsMinimized
5.2-6 AlphabetInvolution
5.2-7 IsBireversible
5.2-8 StateGrowth
5.2-9 Degree
5.2-10 IsFinitaryFRElement
5.2-11 Depth
5.2-12 IsBoundedFRElement
5.2-13 IsPolynomialGrowthFRElement
5.2-14 Signatures
5.2-15 VertexTransformationsFRMachine
5.2-16 FixedRay
5.2-17 IsLevelTransitive
5.2-18 AsMealyMachine
5.2-19 AsMealyMachine
5.2-20 AsMealyElement
5.2-21 AsIntMealyMachine
5.2-22 TopElement
5.2-23 ConfinalityClasses
5.2-24 Germs
5.2-25 HasOpenSetConditionFRElement
5.2-26 LimitMachine
5.2-27 NucleusMachine
5.2-28 GuessMealyElement
6 Linear machines and elements
6.1 Methods and operations for `LinearFRMachine`s and `LinearFRElement`s
6.1-1 VectorMachine
6.1-2 AlgebraMachine
6.1-3 Transition
6.1-4 Transitions
6.1-5 NestedMatrixState
6.1-6 ActivitySparse
6.1-7 Activities
6.1-8 IsConvergent
6.1-9 TransposedFRElement
6.1-10 LDUDecompositionFRElement
6.1-11 GuessVectorElement
6.1-12 AsLinearMachine
6.1-13 AsVectorMachine
6.1-14 AsAlgebraMachine
6.1-15 AsVectorMachine
6.1-16 AsAlgebraMachine
7 Self-similar groups, monoids and semigroups
7.1 Creators for FR semigroups
7.1-1 FRGroup
7.1-2 SCGroup
7.1-3 Correspondence
7.1-4 FullSCGroup
7.1-5 FRMachineFRGroup
7.1-6 IsomorphismFRGroup
7.1-7 IsomorphismMealyGroup
7.1-8 FRGroupByVirtualEndomorphism
7.1-9 TreeWreathProduct
7.1-10 WeaklyBranchedEmbedding
7.2 Operations for FR semigroups
7.2-1 PermGroup
7.2-2 PcGroup
7.2-3 TransMonoid
7.2-4 TransSemigroup
7.2-5 EpimorphismGermGroup
7.2-6 StabilizerImage
7.2-7 LevelStabilizer
7.2-8 IsStateClosedFRSemigroup
7.2-9 StateClosure
7.2-10 IsRecurrentFRSemigroup
7.2-11 IsLevelTransitive
7.2-12 IsInfinitelyTransitive
7.2-13 IsFinitaryFRSemigroup
7.2-14 Degree
7.2-15 HasOpenSetConditionFRSemigroup
7.2-16 IsContracting
7.2-17 NucleusOfFRSemigroup
7.2-18 NucleusMachine
7.2-19 BranchingSubgroup
7.2-20 FindBranchingSubgroup
7.2-21 IsBranched
7.2-22 IsBranchingSubgroup
7.2-23 TopVertexTransformations
7.2-24 VertexTransformations
7.2-25 VirtualEndomorphism
7.2-26 EpimorphismFromFpGroup
7.2-27 IsomorphismSubgroupFpGroup
7.3 Properties for infinite groups
7.3-1 IsTorsionGroup
7.3-2 IsTorsionFreeGroup
7.3-3 IsAmenableGroup
7.3-4 IsVirtuallySimpleGroup
7.3-5 IsResiduallyFinite
7.3-6 IsSQUniversal
7.3-7 IsJustInfinite
8 Algebras
8.1 Creators for FR algebras
8.1-1 FRAlgebra
8.1-2 SCAlgebra
8.2 Operations for FR algebras
8.2-1 MatrixQuotient
8.2-2 ThinnedAlgebra
9 Iterated monodromy groups
9.1 Creators and operations for IMG FR machines
9.1-1 IMGFRMachine
9.1-2 IMGRelator
9.1-3 Mating
9.1-4 PolynomialFRMachine
9.1-5 DBRationalIMGGroup
9.1-6 ValueRational
9.1-7 CriticalValuesQuadraticRational
9.1-8 CanonicalQuadraticRational
9.1-9 PostCriticalMachine
9.2 Spiders
9.2-1 RationalFunction
9.2-2 IMGFRMachine
10 Examples
10.1 Examples of groups
10.1-1 FullBinaryGroup
10.1-2 BinaryKneadingGroup
10.1-3 BasilicaGroup
10.1-4 AddingGroup
10.1-5 BinaryAddingGroup
10.1-6 MixerGroup
10.1-7 SunicGroup
10.1-8 GrigorchukMachines
10.1-9 GrigorchukMachine
10.1-10 GrigorchukOverGroup
10.1-11 GrigorchukEvilTwin
10.1-12 BrunnerSidkiVieiraGroup
10.1-13 AleshinGroups
10.1-14 AleshinGroup
10.1-15 BabyAleshinGroup
10.1-16 SidkiFreeGroup
10.1-17 GuptaSidkiGroups
10.1-18 GuptaSidkiGroup
10.1-19 NeumannGroup
10.1-20 FabrykowskiGuptaGroup
10.1-21 OtherSpinalGroup
10.1-22 GammaPQMachine
10.1-23 HanoiGroup
10.1-24 DahmaniGroup
10.1-25 MamaghaniGroup
10.1-26 WeierstrassGroup
10.1-27 FRAffineGroup
10.1-28 CayleyGroup
10.2 Examples of semigroups
10.2-1 I2Machine
10.2-2 I4Machine
10.3 Bacher's determinant identities
10.4 VH groups
10.4-1 VHStructure
10.4-2 VerticalAction
10.4-3 VHGroup
10.4-4 IsIrreducibleVHGroup
10.4-5 MaximalSimpleSubgroup
11 FR implementation details
11.1 The family of FR objects
11.1-1 FRMFamily
11.1-2 FREFamily
11.1-3 AlphabetOfFRObject
11.1-4 AsPermutation
11.1-5 AsTransformation
11.2 Filters for `FRObject`s
11.2-1 IsGroupFRMachine
11.2-2 IsFRMachineStrRep
11.2-3 IsMealyMachine
11.2-4 IsMealyElement
11.2-5 IsMealyMachineIntRep
11.2-6 IsMealyMachineDomainRep
11.2-7 IsVectorFRMachineRep
11.2-8 IsAlgebraFRMachineRep
11.2-9 IsLinearFRMachine
11.2-10 IsLinearFRElement
11.2-11 IsFRElement
11.2-12 IsFRObject
11.2-13 IsFRMachine
11.2-14 IsInvertible
11.2-15 IsFRGroup
11.2-16 IsFRAlgebra
11.3 Some of the algorithms implemented
11.3-1 FRMachineRWS
11.3-2 Order of FR elements
11.3-3 Membership in semigroups
11.3-4 Order of groups
11.3-5 Images and preimages of some groups in f.p. and l.p. groups
11.3-6 Comparison of FR, Mealy, vector, and algebra elements
11.3-7 Inverses of linear elements
12 Miscellanea
12.1 Helpers
12.1-1 maybe
12.1-2 ReturnMaybe
12.1-3 TensorSum
12.1-4 TensorProductX
12.1-5 DirectSum
12.1-6 PeriodicList
12.1-7 CompressPeriodicList
12.1-8 IsConfinal
12.1-9 ConfinalityClass
12.1-10 LargestCommonPrefix
12.1-11 WordGrowth
12.1-12 ShortGroupRelations
12.1-13 ShortGroupWordInSet
12.1-14 SurfaceBraidFpGroup
12.1-15 CharneyBraidFpGroup
12.1-16 ArtinRepresentation
12.1-17 StringByInt
12.1-18 PositionTower
12.1-19 CoefficientsInAbelianExtension
12.1-20 MagmaEndomorphismByImagesNC
12.1-21 MagmaHomomorphismByImagesNC
12.1-22 NewFIFO
12.1-23 ProductIdeal
12.1-24 DimensionSeries
12.1-25 Trans
12.2 User settings
12.2-1 InfoFR
12.2-2 FR_SEARCH
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