  
  
                                    [1X [5XUGALY[105X [101X
  
  
                       [1X Universal Groups Acting LocallY [101X
  
  
                                     4.1.3
  
  
                                  7 July 2023
  
  
                                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;0YSection  [14X4.6[114X  is  due  to Tasman Fell. The 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
    4.6 [33X[0;0YPGL₂ over the p-adic numbers[133X
      4.6-1 LocalActionPGL2Qp
      4.6-2 LocalActionPSL2Qp
  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
