  
  [1X5 [33X[0;0YDiscreteness[133X[101X
  
  [33X[0;0YThis  chapter  contains  functions  that  are  related  to  the discreteness
  property (D) presented in Proposition 3.12 of [Tor20].[133X
  
  
  [1X5.1 [33X[0;0YThe discreteness condition (D)[133X[101X
  
  [33X[0;0YSaid proposition shows that for a given [23XF\le \mathrm{Aut}(B_{d,k})[123X the group
  [23X\mathrm{U}_{k}(F)[123X is discrete if and only if the maximal compatible subgroup
  [23XC(F)[123X of [23XF[123X satisfies condition (D):[133X
  
  
  [24X[33X[0;6Y\forall \omega \in \Omega: F_{T_{\omega}}=\{\mathrm{id}\},[133X
  
  [124X
  
  [33X[0;0Ywhere  [23XT_{\omega}[123X is the [23Xk-1[123X-neighbourhood of the edge [23X(b,b_{\omega})[123X inside
  [23XB_{d,k}[123X.  In  other  words, [23XF[123X satisfies (D) if and only if the compatibility
  set  [23XC_{F}(\mathrm{id},\omega)=\{\mathrm{id}\}[123X.  We  distinguish  between  [23XF[123X
  satisfying  condition  (D)  and  [23X\mathrm{U}_{k}(F)[123X  being  discrete with the
  methods [2XSatisfiesD[102X ([14X5.2-1[114X) and [2XYieldsDiscreteUniversalGroup[102X ([14X5.2-2[114X) below.[133X
  
  
  [1X5.2 [33X[0;0YDiscreteness[133X[101X
  
  [1X5.2-1 SatisfiesD[101X
  
  [33X[1;0Y[29X[2XSatisfiesD[102X( [3XF[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X  if  [3XF[103X  satisfies  the  discreteness condition (D), and [9Xfalse[109X
            otherwise.[133X
  
  [33X[0;0YThe  argument of this attribute is a local action [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X
  (see [2XIsLocalAction[102X ([14X2.1-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=LocalActionGamma(3,SymmetricGroup(3));[127X[104X
    [4X[28XGroup([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XSatisfiesD(G);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X5.2-2 YieldsDiscreteUniversalGroup[101X
  
  [33X[1;0Y[29X[2XYieldsDiscreteUniversalGroup[102X( [3XF[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if [23X\mathrm{U}_{k}(F)[123X is discrete, and [9Xfalse[109X otherwise.[133X
  
  [33X[0;0YThe  argument of this attribute is a local action [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X
  (see [2XIsLocalAction[102X ([14X2.1-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=LocalActionGamma(3,SymmetricGroup(3));[127X[104X
    [4X[28XGroup([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XYieldsDiscreteUniversalGroup(G);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=LocalAction(3,2,Group((1,2)));[127X[104X
    [4X[28XGroup([ (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27XYieldsDiscreteUniversalGroup(F);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSatisfiesD(F);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XC:=MaximalCompatibleSubgroup(F);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XSatisfiesD(C);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X5.3 [33X[0;0YCocycles[133X[101X
  
  [33X[0;0YSubgroups  [23XF\le\mathrm{Aut}(B_{d,k})[123X  that satisfy both (C) and (D) admit an
  involutive  compatibility  cocycle,  i.e. a map [23Xz:F\times\{1,\ldots,d\}\to F[123X
  that  satisfies  certain  properties,  see  [Tor20,  Section  3.2.2]. When [23XF[123X
  satisfies  just (C), it may still admit an involutive compatibility cocycle.
  In  this  case,  F admits an extension [23X\Gamma_{z}(F)\le\mathrm{Aut}(B_{d,k})[123X
  that  satisfies  both  (C) and (D). Involutive compatibility cocycles can be
  searched    for    using    [2XInvolutiveCompatibilityCocycle[102X    ([14X5.3-1[114X)    and
  [2XAllInvolutiveCompatibilityCocycles[102X ([14X5.3-2[114X) below.[133X
  
  [1X5.3-1 InvolutiveCompatibilityCocycle[101X
  
  [33X[1;0Y[29X[2XInvolutiveCompatibilityCocycle[102X( [3XF[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Yan  involutive  compatibility  cocycle  of  [3XF[103X,  which is a mapping
            [3XF[103X[23X\times[123X[10X[1..d][110X[23X\to[123X[3XF[103X  with certain properties, if it exists, and [9Xfail[109X
            otherwise. When [3Xk[103X [23X=1[123X, the standard cocycle is returned.[133X
  
  [33X[0;0YThe  argument of this attribute is a local action [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X
  (see [2XIsLocalAction[102X ([14X2.1-1[114X)), which is compatible (see [2XSatisfiesC[102X ([14X3.3-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=LocalAction(3,1,AlternatingGroup(3));;[127X[104X
    [4X[25Xgap>[125X [27Xz:=InvolutiveCompatibilityCocycle(F);;[127X[104X
    [4X[25Xgap>[125X [27Xmt:=RandomSource(IsMersenneTwister,1);;[127X[104X
    [4X[25Xgap>[125X [27Xa:=Random(mt,F);; dir:=Random(mt,[1..3]);;[127X[104X
    [4X[25Xgap>[125X [27Xa; Image(z,[a,dir]);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[28X(1,2,3)[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=LocalActionGamma(3,AlternatingGroup(3));[127X[104X
    [4X[28XGroup([ (1,4,5)(2,3,6) ])[128X[104X
    [4X[25Xgap>[125X [27XInvolutiveCompatibilityCocycle(G) <> fail;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XInvolutiveCompatibilityCocycle(AutBall(3,2));[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X5.3-2 AllInvolutiveCompatibilityCocycles[101X
  
  [33X[1;0Y[29X[2XAllInvolutiveCompatibilityCocycles[102X( [3XF[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe list of all involutive compatibility cocycles of [23XF[123X.[133X
  
  [33X[0;0YThe  argument of this attribute is a local action [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X
  (see [2XIsLocalAction[102X ([14X2.1-1[114X)), which is compatible (see [2XSatisfiesC[102X ([14X3.3-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS3:=LocalAction(3,1,SymmetricGroup(3));;[127X[104X
    [4X[25Xgap>[125X [27XSize(AllInvolutiveCompatibilityCocycles(S3));[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XSize(AllInvolutiveCompatibilityCocycles(LocalActionGamma(3,SymmetricGroup(3))));[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
