  
  
                         [1XFunctionally recursive groups[101X
  
  
                              [1XSelf-similar groups[101X
  
  
                                 Version 2.4.10
  
  
                                   10/08/2022
  
  
                               Laurent Bartholdi
  
  
  
            [33X[0;10YGroups  generated  by  automata or satisfying functional
            recursions[133X
  
  
  
  Laurent Bartholdi
      Email:    [7Xmailto:laurent.bartholdi@gmail.com[107X
      Homepage: [7Xhttps://www.math.uni-sb.de/ag/bartholdi/[107X
  
  
  Address: [33X[0;9YMathematisches Institut[133X
           [33X[0;9YBunsenstraße 3-5[133X
           [33X[0;9YD-37073 Göttingen[133X
           [33X[0;9YGermany[133X
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThis  document  describes  the package [5XFR[105X, which implements in [5XGAP[105X the basic
  objects of Mealy machines and functional recursions; and handles groups that
  they generate.[133X
  
  [33X[0;0YGroups  defined by a recursive action on a rooted tree can be defined in [5XGAP[105X
  via  their recursion. Various algorithms are implemented to manipulate these
  groups and their elements.[133X
  
  [33X[0;0YFor  comments  or questions on [5XFR[105X please contact the author; this package is
  still under development.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2006-2012 by Laurent Bartholdi[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YPart  of  this  work  is/was  supported  by  the  "Swiss  National  Fund for
  Scientific Research" and the "German Science Foundation".[133X
  
  
  -------------------------------------------------------
  [1XColophon[101X
  [33X[0;0YThis  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.[133X
  
  [33X[0;0YThe  first routines I wrote constructed finite representations of the groups
  considered,  so  as  to  get  insight from fast calculations within [5XGAP[105X. 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.[133X
  
  [33X[0;0YAround  2000,  I  had developed quite a few routines, which I assembled in a
  [5XGAP[105X  package,  that  dealt  directly  with infinite groups. This package was
  primitive at its core, but was extended with various routines as they became
  useful.[133X
  
  [33X[0;0YI  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.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (FR)[101X
  
  1 [33X[0;0YLicensing[133X
  2 [33X[0;0YFR package[133X
    2.1 [33X[0;0YA brief mathematical introduction[133X
    2.2 [33X[0;0YAn example session[133X
  3 [33X[0;0YFunctionally recursive machines[133X
    3.1 [33X[0;0YTypes of machines[133X
    3.2 [33X[0;0YProducts of machines[133X
    3.3 [33X[0;0YCreators for [10XFRMachine[110Xs[133X
      3.3-1 FRMachineNC
      3.3-2 FRMachine
      3.3-3 UnderlyingFRMachine
      3.3-4 AsGroupFRMachine
      3.3-5 AsGroupFRMachine
    3.4 [33X[0;0YAttributes for [10XFRMachine[110Xs[133X
      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 [33X[0;0YOperations for [10XFRMachine[110Xs[133X
      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 [33X[0;0YFunctionally recursive elements[133X
    4.1 [33X[0;0YCreators for [10XFRElement[110Xs[133X
      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 [33X[0;0YOperations and Attributes for [10XFRElement[110Xs[133X
      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 [33X[0;0YMealy machines and elements[133X
    5.1 [33X[0;0YCreators for [10XMealyMachine[110Xs and [10XMealyElement[110Xs[133X
      5.1-1 MealyMachine
      5.1-2 MealyMachine
      5.1-3 MealyMachineNC
      5.1-4 AllMealyMachines
    5.2 [33X[0;0YOperations and Attributes for [10XMealyMachine[110Xs and [10XMealyElement[110Xs[133X
      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 IsLevelTransitiveFRElement
      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 [33X[0;0YLinear machines and elements[133X
    6.1 [33X[0;0YMethods and operations for [10XLinearFRMachine[110Xs and [10XLinearFRElement[110Xs[133X
      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 [33X[0;0YSelf-similar groups, monoids and semigroups[133X
    7.1 [33X[0;0YCreators for FR semigroups[133X
      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 [33X[0;0YOperations for FR semigroups[133X
      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 IsLevelTransitiveFRGroup
      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 [33X[0;0YProperties for infinite groups[133X
      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 [33X[0;0YAlgebras[133X
    8.1 [33X[0;0YCreators for FR algebras[133X
      8.1-1 FRAlgebra
      8.1-2 SCAlgebra
      8.1-3 NucleusOfFRAlgebra
      8.1-4 BranchingIdeal
    8.2 [33X[0;0YOperations for FR algebras[133X
      8.2-1 MatrixQuotient
      8.2-2 ThinnedAlgebra
      8.2-3 Nillity
      8.2-4 DegreeOfHomogeneousElement
  9 [33X[0;0YExamples[133X
    9.1 [33X[0;0YExamples of groups[133X
      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 [33X[0;0YExamples of semigroups[133X
      9.2-1 I2Machine
      9.2-2 I4Machine
    9.3 [33X[0;0YExamples of algebras[133X
      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 [33X[0;0YBacher's determinant identities[133X
    9.5 [33X[0;0YVH groups[133X
      9.5-1 VHStructure
      9.5-2 VerticalAction
      9.5-3 VHGroup
      9.5-4 IsIrreducibleVHGroup
      9.5-5 MaximalSimpleSubgroup
  10 [33X[0;0YFR implementation details[133X
    10.1 [33X[0;0YThe family of FR objects[133X
      10.1-1 FRMFamily
      10.1-2 FREFamily
      10.1-3 AlphabetOfFRObject
      10.1-4 AsPermutation
      10.1-5 AsTransformation
    10.2 [33X[0;0YFilters for [10XFRObject[110Xs[133X
      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 [33X[0;0YSome of the algorithms implemented[133X
      10.3-1 FRMachineRWS
      10.3-2 [33X[0;0YOrder of FR elements[133X
      10.3-3 [33X[0;0YMembership in semigroups[133X
      10.3-4 [33X[0;0YThe conjugacy problem[133X
      10.3-5 OrbitSignalizer
      10.3-6 FRConjugacyAlgorithm
      10.3-7 FRBranchGroupConjugacyData
      10.3-8 [33X[0;0YOrder of groups[133X
      10.3-9 [33X[0;0YImages and preimages of some groups in f.p. and l.p. groups[133X
      10.3-10 [33X[0;0YComparison of FR, Mealy, vector, and algebra elements[133X
      10.3-11 [33X[0;0YInverses of linear elements[133X
  11 [33X[0;0YMiscellanea[133X
    11.1 [33X[0;0YGeneric operations[133X
      11.1-1 TensorSum
      11.1-2 TensorProduct
      11.1-3 DirectSum
    11.2 [33X[0;0YPeriodic lists[133X
      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 [33X[0;0YWord growth[133X
      11.3-1 WordGrowth
    11.4 [33X[0;0YFinding short relations[133X
      11.4-1 ShortGroupRelations
      11.4-2 ShortGroupWordInSet
    11.5 [33X[0;0YBraid groups[133X
      11.5-1 SurfaceBraidFpGroup
      11.5-2 CharneyBraidFpGroup
      11.5-3 ArtinRepresentation
    11.6 [33X[0;0YDirichlet series[133X
      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 [33X[0;0YProjective representations[133X
      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 [33X[0;0YMiscellanea[133X
      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 [33X[0;0YUser settings[133X
      11.9-1 InfoFR
      11.9-2 SEARCH@
  
  
  [32X
