  
  
                                    [1X [5XUGALY[105X [101X
  
  
                       [1X Universal Groups Acting LocallY [101X
  
  
                                     4.0.3
  
  
                                  14 July 2022
  
  
                                Khalil Hannouch
  
                                Stephan Tornier
  
  
  
  Khalil Hannouch
      Email:    [7Xmailto:khalil.hannouch@newcastle.edu.au[107X
      Homepage: [7Xhttps://www.newcastle.edu.au/profile/khalil-hannouch[107X
      Address:  [33X[0;14YKhalil Hannouch[133X
                [33X[0;14YThe University of Newcastle[133X
                [33X[0;14YSchool of Information and Physical Sciences[133X
                [33X[0;14YUniversity Drive[133X
                [33X[0;14Y2308 Callaghan NSW[133X
                [33X[0;14YAustralia[133X
  
  
  Stephan Tornier
      Email:    [7Xmailto:stephan.tornier@newcastle.edu.au[107X
      Homepage: [7Xhttps://www.newcastle.edu.au/profile/stephan-tornier[107X
      Address:  [33X[0;14YStephan Tornier[133X
                [33X[0;14YThe University of Newcastle[133X
                [33X[0;14YSchool of Information and Physical Sciences[133X
                [33X[0;14YUniversity Drive[133X
                [33X[0;14Y2308 Callaghan NSW[133X
                [33X[0;14YAustralia[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0Y[5XUGALY[105X  ([12XU[112Xniversal  [12XG[112Xroups  [12XA[112Xcting  [12XL[112Xocall[12XY[112X)  is  a [12XGAP[112X package that provides
  methods  to  create, analyse and find local actions of generalised universal
  groups  acting  on  locally finite regular trees, following Burger-Mozes and
  Tornier.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y[5XUGALY[105X  is  free software; you can redistribute it and/or modify it under the
  terms        of       the       GNU       General       Public       License
  ([7Xhttps://www.fsf.org/licenses/gpl.html[107X)  as  published  by the Free Software
  Foundation;  either  version 3 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe  second  author  owes  thanks to Marc Burger and George Willis for their
  support and acknowledges contributions from the SNSF Doc.Mobility fellowship
  172120  and  the  ARC Discovery Project DP120100996 to the development of an
  early  version  of  this  codebase.  In its present form, the development of
  [5XUGALY[105X  was  made possible by the ARC Laureate Fellowship FL170100032 and the
  ARC  DECRA  Fellowship  DE210100180.  Finally,  we  owe  thanks  to  Laurent
  Bartholdi  for  guiding  us through a reviewing process that has resulted in
  substantial  improvements,  and to Max Horn for helping with a documentation
  issue.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (UGALY)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YPurpose[133X
  2 [33X[0;0YPreliminaries[133X
    2.1 [33X[0;0YLocal actions[133X
      2.1-1 IsLocalAction
      2.1-2 LocalAction
      2.1-3 LocalActionNC
      2.1-4 LocalActionDegree
      2.1-5 LocalActionRadius
      2.1-6 LocalAction
      2.1-7 Projection
      2.1-8 ImageOfProjection
    2.2 [33X[0;0YFinite balls[133X
      2.2-1 AutBall
    2.3 [33X[0;0YAddresses and leaves[133X
      2.3-1 BallAddresses
      2.3-2 LeafAddresses
      2.3-3 AddressOfLeaf
      2.3-4 LeafOfAddress
      2.3-5 ImageAddress
      2.3-6 ComposeAddresses
  3 [33X[0;0YCompatibility[133X
    3.1 [33X[0;0YThe compatibility condition (C)[133X
    3.2 [33X[0;0YCompatible elements[133X
      3.2-1 AreCompatibleBallElements
      3.2-2 CompatibleBallElement
      3.2-3 [33X[0;0YCompatibilitySet[133X
      3.2-4 AssembleAutomorphism
    3.3 [33X[0;0YCompatible subgroups[133X
      3.3-1 MaximalCompatibleSubgroup
      3.3-2 SatisfiesC
      3.3-3 CompatibleSubgroups
      3.3-4 ConjugacyClassRepsCompatibleSubgroups
      3.3-5 ConjugacyClassRepsCompatibleGroupsWithProjection
  4 [33X[0;0YExamples[133X
    4.1 [33X[0;0YDiscrete groups[133X
      4.1-1 [33X[0;0YLocalActionElement[133X
      4.1-2 [33X[0;0YLocalActionGamma[133X
      4.1-3 [33X[0;0YLocalActionDelta[133X
    4.2 [33X[0;0YMaximal extensions[133X
      4.2-1 [33X[0;0YLocalActionPhi[133X
    4.3 [33X[0;0YNormal subgroups and partitions[133X
      4.3-1 [33X[0;0YLocalActionPhi[133X
    4.4 [33X[0;0YAbelian quotients[133X
      4.4-1 SignHomomorphism
      4.4-2 AbelianizationHomomorphism
      4.4-3 SpheresProduct
      4.4-4 LocalActionPi
    4.5 [33X[0;0YSemidirect products[133X
      4.5-1 [33X[0;0YCompatibleKernels[133X
      4.5-2 [33X[0;0YLocalActionSigma[133X
  5 [33X[0;0YDiscreteness[133X
    5.1 [33X[0;0YThe discreteness condition (D)[133X
    5.2 [33X[0;0YDiscreteness[133X
      5.2-1 SatisfiesD
      5.2-2 YieldsDiscreteUniversalGroup
    5.3 [33X[0;0YCocycles[133X
      5.3-1 InvolutiveCompatibilityCocycle
      5.3-2 AllInvolutiveCompatibilityCocycles
  
  
  [32X
