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# Functionally recursive groups

## Self-similar groups

Version 2.3.3

27/02/2016

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
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://laurentbartholdi.github.com/fr/manual.pdf in PDF format, and may be accessed online at http://laurentbartholdi.github.com/fr/.

The latest release of the package may be downloaded as https://github.com/laurentbartholdi/fr/archive/2.3.3.tar.gz (tar, gzipped) and https://github.com/laurentbartholdi/fr/archive/2.3.3.zip (zip). The latest repository version (possibly unstable) may be downloaded as https://github.com/laurentbartholdi/fr/tarball/master (tar, gzipped), https://github.com/laurentbartholdi/fr.git (git repository), or explored at https://github.com/laurentbartholdi/fr/tree/master/.

The last source compatible with GAP 4.4.12 may be downloaded as https://github.com/downloads/laurentbartholdi/fr/fr-1.1.2.tar.gz (tar, gzipped).

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

© 2006-2012 by Laurent Bartholdi

### Acknowledgements

Part of this work is/was supported by the "Swiss National Fund for Scientific Research" and the "German Science Foundation".

### 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 AsGroupFRMachine
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 Transitions
3.4-6 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 ChangeFRMachineBasis
3.5-11 Minimized
3.5-12 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 Transitions
4.2-5 Portrait
4.2-6 DecompositionOfFRElement
4.2-7 StateSet
4.2-8 State
4.2-9 States
4.2-10 FixedStates
4.2-11 LimitStates
4.2-12 IsFiniteStateFRElement
4.2-13 NucleusOfFRMachine
4.2-14 InitialState
4.2-15 \^
4.2-16 \*
4.2-17 \[\]
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 LimitFRMachine
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 AssociativeObject
6.1-3 AlgebraMachine
6.1-4 Transition
6.1-5 Transitions
6.1-6 NestedMatrixState
6.1-7 ActivitySparse
6.1-8 Activities
6.1-9 IsConvergent
6.1-10 TransposedFRElement
6.1-11 LDUDecompositionFRElement
6.1-12 GuessVectorElement
6.1-13 AsLinearMachine
6.1-14 AsVectorMachine
6.1-15 AsAlgebraMachine
6.1-16 AsVectorMachine
6.1-17 AsAlgebraMachine
7 Self-similar groups, monoids and semigroups
7.1 Creators for FR semigroups
7.1-1 FRGroup
7.1-2 NewSemigroupFRMachine
7.1-3 SCGroup
7.1-4 Correspondence
7.1-5 FullSCGroup
7.1-6 FRMachineFRGroup
7.1-7 IsomorphismFRGroup
7.1-8 IsomorphismMealyGroup
7.1-9 FRGroupByVirtualEndomorphism
7.1-10 TreeWreathProduct
7.1-11 WeaklyBranchedEmbedding
7.2 Operations for FR semigroups
7.2-1 PermGroup
7.2-2 PcGroup
7.2-3 TransformationMonoid
7.2-4 TransformationSemigroup
7.2-5 EpimorphismGermGroup
7.2-6 GermData
7.2-7 StabilizerImage
7.2-8 LevelStabilizer
7.2-9 IsStateClosed
7.2-10 StateClosure
7.2-11 IsRecurrentFRSemigroup
7.2-12 IsLevelTransitive
7.2-13 IsInfinitelyTransitive
7.2-14 IsFinitaryFRSemigroup
7.2-15 Degree
7.2-16 HasOpenSetConditionFRSemigroup
7.2-17 HasCongruenceProperty
7.2-18 IsContracting
7.2-19 NucleusOfFRSemigroup
7.2-20 NucleusMachine
7.2-21 AdjacencyBasesWithOne
7.2-22 BranchingSubgroup
7.2-23 FindBranchingSubgroup
7.2-24 IsBranched
7.2-25 IsBranchingSubgroup
7.2-26 BranchStructure
7.2-27 TopVertexTransformations
7.2-28 VertexTransformations
7.2-29 VirtualEndomorphism
7.2-30 EpimorphismFromFpGroup
7.2-31 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
8.1-3 NucleusOfFRAlgebra
8.1-4 BranchingIdeal
8.2 Operations for FR algebras
8.2-1 MatrixQuotient
8.2-2 ThinnedAlgebra
8.2-3 Nillity
8.2-4 DegreeOfHomogeneousElement
9 Examples
9.1 Examples of groups
9.1-1 FullBinaryGroup
9.1-2 BinaryKneadingGroup
9.1-3 BasilicaGroup
9.1-4 FornaessSibonyGroup
9.1-5 AddingGroup
9.1-6 BinaryAddingGroup
9.1-7 MixerGroup
9.1-8 SunicGroup
9.1-9 GrigorchukMachines
9.1-10 GrigorchukMachine
9.1-11 GrigorchukOverGroup
9.1-12 GrigorchukTwistedTwin
9.1-13 BrunnerSidkiVieiraGroup
9.1-14 AleshinGroups
9.1-15 AleshinGroup
9.1-16 BabyAleshinGroup
9.1-17 SidkiFreeGroup
9.1-18 GuptaSidkiGroups
9.1-19 GuptaSidkiGroup
9.1-20 NeumannGroup
9.1-21 FabrykowskiGuptaGroup
9.1-22 ZugadiSpinalGroup
9.1-23 GammaPQMachine
9.1-24 RattaggiGroup
9.1-25 HanoiGroup
9.1-26 DahmaniGroup
9.1-27 MamaghaniGroup
9.1-28 WeierstrassGroup
9.1-29 StrichartzGroup
9.1-30 FRAffineGroup
9.1-31 CayleyGroup
9.2 Examples of semigroups
9.2-1 I2Machine
9.2-2 I4Machine
9.3 Examples of algebras
9.3-1 PSZAlgebra
9.3-2 GrigorchukThinnedAlgebra
9.3-3 GuptaSidkiThinnedAlgebra
9.3-4 GrigorchukLieAlgebra
9.3-5 SidkiFreeAlgebra
9.3-6 SidkiMonomialAlgebra
9.4 Bacher's determinant identities
9.5 VH groups
9.5-1 VHStructure
9.5-2 VerticalAction
9.5-3 VHGroup
9.5-4 IsIrreducibleVHGroup
9.5-5 MaximalSimpleSubgroup
10 FR implementation details
10.1 The family of FR objects
10.1-1 FRMFamily
10.1-2 FREFamily
10.1-3 AlphabetOfFRObject
10.1-4 AsPermutation
10.1-5 AsTransformation
10.2 Filters for `FRObject`s
10.2-1 IsGroupFRMachine
10.2-2 IsFRMachineStrRep
10.2-3 IsMealyMachine
10.2-4 IsMealyElement
10.2-5 IsMealyMachineIntRep
10.2-6 IsMealyMachineDomainRep
10.2-7 IsVectorFRMachineRep
10.2-8 IsAlgebraFRMachineRep
10.2-9 IsLinearFRMachine
10.2-10 IsLinearFRElement
10.2-11 IsFRElement
10.2-12 IsFRMealyElement
10.2-13 IsFRObject
10.2-14 IsFRMachine
10.2-15 IsInvertible
10.2-16 IsFRGroup
10.2-17 IsFRAlgebra
10.3 Some of the algorithms implemented
10.3-1 FRMachineRWS
10.3-2 Order of FR elements
10.3-3 Membership in semigroups
10.3-4 The conjugacy problem
10.3-5 OrbitSignalizer
10.3-6 FRConjugacyAlgorithm
10.3-7 FRBranchGroupConjugacyData
10.3-8 Order of groups
10.3-9 Images and preimages of some groups in f.p. and l.p. groups
10.3-10 Comparison of FR, Mealy, vector, and algebra elements
10.3-11 Inverses of linear elements
11 Miscellanea
11.1 Generic operations
11.1-1 TensorSum
11.1-2 TensorProduct
11.1-3 DirectSum
11.2 Periodic lists
11.2-1 PeriodicListsFamily
11.2-2 PeriodicList
11.2-3 CompressPeriodicList
11.2-4 IsConfinal
11.2-5 ConfinalityClass
11.2-6 LargestCommonPrefix
11.3 Word growth
11.3-1 WordGrowth
11.4 Finding short relations
11.4-1 ShortGroupRelations
11.4-2 ShortGroupWordInSet
11.5 Braid groups
11.5-1 SurfaceBraidFpGroup
11.5-2 CharneyBraidFpGroup
11.5-3 ArtinRepresentation
11.6 Dirichlet series
11.6-1 DirichletSeries
11.6-2 DegreeDirichletSeries
11.6-3 SpreadDirichletSeries
11.6-4 ShiftDirichletSeries
11.6-5 ShrunkDirichletSeries
11.6-6 ZetaSeriesOfGroup
11.6-7 ValueOfDirichletSeries
11.7 Projective representations
11.7-1 IsProjectiveRepresentation
11.7-2 ProjectiveRepresentationByFunction
11.7-3 LinearRepresentationByImages
11.7-4 DegreeOfProjectiveRepresentation
11.7-5 ProjectiveExtension
11.7-6 ProjectiveQuotient
11.8 Miscellanea
11.8-1 ForwardOrbit
11.8-2 StringByInt
11.8-3 PositionInTower
11.8-4 RenameSubobjects
11.8-5 CoefficientsInAbelianExtension
11.8-6 MagmaEndomorphismByImagesNC
11.8-7 MagmaHomomorphismByImagesNC
11.8-8 Draw
11.8-9 IsFIFO
11.8-10 ProductIdeal
11.8-11 DimensionSeries
11.8-12 AlgebraHomomorphismByFunction
11.8-13 IsFpLieAlgebra
11.8-14 JenningsLieAlgebra
11.8-15 SolutionMatModN
11.8-16 SolutionMatMod1
11.8-17 CyclotomicByArgument
11.8-18 ArgumentOfCyclotomic
11.9 User settings
11.9-1 InfoFR
11.9-2 SEARCH@
References
Index
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