  
  [1X2 [33X[0;0YPreliminaries[133X[101X
  
  [33X[0;0YWe  recall  the  following notation from the Introduction which is essential
  throughout  this  manual,  cf.  [Tor20].  Let [23X\Omega[123X be a set of cardinality
  [23Xd\in\mathbb{N}_{\ge  3}[123X  and  let  [23XT_{d}=(V,E)[123X  denote  the  [23Xd[123X-regular tree,
  following  the graph theory notation in [Ser80]. A [13Xlabelling[113X [23Xl[123X of [23XT_{d}[123X is a
  map   [23Xl:E\to\Omega[123X   such   that   for   every   [23Xx\in   V[123X   the  restriction
  [23Xl_{x}:E(x)\to\Omega,\ e\mapsto l(e)[123X is a bijection, and [23Xl(e)=l(\overline{e})[123X
  for  all  [23Xe\in  E[123X.  For  every  [23Xk\in\mathbb{N}[123X,  fix a tree [23XB_{d,k}[123X which is
  isomorphic to a ball of radius [23Xk[123X around a vertex in [23XT_{d}[123X and carry over the
  labelling  of  [23XT_{d}[123X  to  [23XB_{d,k}[123X  via the chosen isomorphism. We denote the
  center of [23XB_{d,k}[123X by [23Xb[123X.[133X
  
  [33X[0;0YFor   every   [23Xx\in   V[123X  there  is  a  unique,  label-respecting  isomorphism
  [23Xl_{x}^{k}:B(x,k)\to     B_{d,k}[123X.    We    define    the    [13X[23Xk[123X-local    action[113X
  [23X\sigma_{k}(g,x)\in\mathrm{Aut}(B_{d,k})[123X       of       an       automorphism
  [23Xg\!\in\!\mathrm{Aut}(T_{d})[123X at a vertex [23Xx\in V[123X via the map[133X
  
  
  [24X[33X[0;6Y\sigma_{k}:\mathrm{Aut}(T_{d})\times              V\to\mathrm{Aut}(B_{d,k}),
  \sigma_{k}(g,x)\mapsto        \sigma_{k}(g,x):=l_{gx}^{k}\circ        g\circ
  (l_{x}^{k})^{-1}.[133X
  
  [124X
  
  
  [1X2.1 [33X[0;0YLocal actions[133X[101X
  
  [33X[0;0YIn  this  package,  local  actions  [23XF\le\mathrm{Aut}(B_{d,k})[123X are handled as
  objects  of  the  category [2XIsLocalAction[102X ([14X2.1-1[114X) and have several attributes
  and  properties introduced throughout this manual. Most importantly, a local
  action  always stores the degree [23Xd[123X and the radius [23Xk[123X of the ball [23XB_{d,k}[123X that
  it acts on.[133X
  
  [1X2.1-1 IsLocalAction[101X
  
  [33X[1;0Y[29X[2XIsLocalAction[102X( [3XF[103X ) [32X filter[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X  if  [23XF[123X  is an object of the category [9XIsLocalAction[109X, and [9Xfalse[109X
            otherwise.[133X
  
  [33X[0;0YLocal  actions  [23XF\le\mathrm{Aut}(B_{d,k})[123X  are  stored  together  with their
  degree   (see  [2XLocalActionDegree[102X  ([14X2.1-4[114X)),  radius  (see  [2XLocalActionRadius[102X
  ([14X2.1-5[114X))  as  well as other attributes and properties in this category. They
  can be initialised using the creator operation [2XLocalAction[102X ([14X2.1-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=WreathProduct(SymmetricGroup(2),SymmetricGroup(3));[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XH:=AutBall(3,2);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(H);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XK:=LocalAction(3,2,G);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(K);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 LocalAction[101X
  
  [33X[1;0Y[29X[2XLocalAction[102X( [3Xd[103X, [3Xk[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  regular  rooted  tree  group  [23XG[123X  as an object of the category
            [2XIsLocalAction[102X  ([14X2.1-1[114X),  checking  that  [3XF[103X is indeed a subgroup of
            [23X\mathrm{Aut}(B_{d,k})[123X.[133X
  
  [33X[0;0YThe  arguments of this method are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X, a radius
  [3Xk[103X [23X\in\mathbb{N}_{0}[123X and a group [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=WreathProduct(SymmetricGroup(2),SymmetricGroup(3));[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(G);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XG:=LocalAction(3,2,G);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsLocalAction(G);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.1-3 LocalActionNC[101X
  
  [33X[1;0Y[29X[2XLocalActionNC[102X( [3Xd[103X, [3Xk[103X, [3XF[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  regular  rooted  tree  group  [23XG[123X  as an object of the category
            [2XIsLocalAction[102X  ([14X2.1-1[114X),  without  checking  that  [3XF[103X  is  indeed  a
            subgroup of [23X\mathrm{Aut}(B_{d,k})[123X.[133X
  
  [33X[0;0YThe  arguments of this method are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X, a radius
  [3Xk[103X [23X\in\mathbb{N}_{0}[123X and a group [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X.[133X
  
  [1X2.1-4 LocalActionDegree[101X
  
  [33X[1;0Y[29X[2XLocalActionDegree[102X( [3XF[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe degree [3Xd[103X of the ball [23XB_{d,k}[123X that [23XF[123X is acting on.[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 [27XA4:=LocalAction(4,1,AlternatingGroup(4));[127X[104X
    [4X[28XAlt( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XF:=LocalActionPhi(3,A4);[127X[104X
    [4X[28X<permutation group with 18 generators>[128X[104X
    [4X[25Xgap>[125X [27XLocalActionDegree(F);[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [1X2.1-5 LocalActionRadius[101X
  
  [33X[1;0Y[29X[2XLocalActionRadius[102X( [3XF[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe radius [3Xk[103X of the ball [23XB_{d,k}[123X that [23XF[123X is acting on.[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 [27XA4:=LocalAction(4,1,AlternatingGroup(4));[127X[104X
    [4X[28XAlt( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XF:=LocalActionPhi(3,A4);[127X[104X
    [4X[28X<permutation group with 18 generators>[128X[104X
    [4X[25Xgap>[125X [27XLocalActionRadius(F);[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [1X2.1-6 LocalAction[101X
  
  [33X[1;0Y[29X[2XLocalAction[102X( [3Xr[103X, [3Xd[103X, [3Xk[103X, [3Xaut[103X, [3Xaddr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe [3Xr[103X-local action [23X\sigma_{r}([123X[3Xaut[103X,[3Xaddr[103X[23X)[123X of the automorphism [3Xaut[103X of
            [23XB_{d,k}[123X at the vertex represented by the address [3Xaddr[103X.[133X
  
  [33X[0;0YThe  arguments  of this method are a radius [3Xr[103X, a degree [3Xd[103X [23X\in\mathbb{N}_{\ge
  3}[123X, a radius [3Xk[103X [23X\in\mathbb{N}[123X, an automorphism [3Xaut[103X of [23XB_{d,k}[123X, and an address
  [3Xaddr[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=(1,3,5)(2,4,6);; a in AutBall(3,2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLocalAction(2,3,2,a,[]);[127X[104X
    [4X[28X(1,3,5)(2,4,6)[128X[104X
    [4X[25Xgap>[125X [27XLocalAction(1,3,2,a,[]);[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27XLocalAction(1,3,2,a,[1]);[127X[104X
    [4X[28X(1,2)[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmt:=RandomSource(IsMersenneTwister,1);;[127X[104X
    [4X[25Xgap>[125X [27Xb:=Random(mt,AutBall(3,4));[127X[104X
    [4X[28X(1,18,11,5,23,14,4,20,10,7,22,16)(2,17,12,6,24,13,3,19,9,8,21,15)[128X[104X
    [4X[25Xgap>[125X [27XLocalAction(2,3,4,b,[3,1]);[127X[104X
    [4X[28X(1,2)(3,6,4,5)[128X[104X
    [4X[25Xgap>[125X [27XLocalAction(3,3,4,b,[3,1]);[127X[104X
    [4X[28XError, the sum of input argument r=3 and the length of input argument[128X[104X
    [4X[28Xaddr=[ 3, 1 ] must not exceed input argument k=4[128X[104X
  [4X[32X[104X
  
  [1X2.1-7 Projection[101X
  
  [33X[1;0Y[29X[2XProjection[102X( [3XF[103X, [3Xr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe       restriction       of       the       projection      map
            [23X\mathrm{Aut}(B_{d,k})\to\mathrm{Aut}(B_{d,r})[123X to [3XF[103X.[133X
  
  [33X[0;0YThe  arguments of this method are a local action [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X,
  and a projection radius [3Xr[103X [23X\le[123X [3Xk[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=LocalActionGamma(4,3,SymmetricGroup(3));[127X[104X
    [4X[28XGroup([ (1,16,19)(2,15,20)(3,13,18)(4,14,17)(5,10,23)(6,9,24)(7,12,22)[128X[104X
    [4X[28X  (8,11,21), (1,9)(2,10)(3,12)(4,11)(5,15)(6,16)(7,13)(8,14)(17,21)(18,22)[128X[104X
    [4X[28X  (19,24)(20,23) ])[128X[104X
    [4X[25Xgap>[125X [27Xpr:=Projection(F,2);[127X[104X
    [4X[28X<action homomorphism>[128X[104X
    [4X[25Xgap>[125X [27Xmt:=RandomSource(IsMersenneTwister,1);;[127X[104X
    [4X[25Xgap>[125X [27Xa:=Random(mt,F);; Image(pr,a);[127X[104X
    [4X[28X(1,2)(3,5)(4,6)[128X[104X
  [4X[32X[104X
  
  [1X2.1-8 ImageOfProjection[101X
  
  [33X[1;0Y[29X[2XImageOfProjection[102X( [3XF[103X, [3Xr[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe local action [23X\sigma_{r}(F,b)\le\mathrm{Aut}(B_{d,r})[123X.[133X
  
  [33X[0;0YThe  arguments of this method are a local action [3XF[103X [23X\le\mathrm{Aut}(B_{d,k})[123X,
  and  a  projection  radius  [3Xr[103X [23X\le[123X [3Xk[103X. This method uses [2XLocalAction[102X ([14X2.1-6[114X) on
  generators rather than [2XProjection[102X ([14X2.1-7[114X) on the group to compute the image.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAutBall(3,2);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XImageOfProjection(AutBall(3,2),1);[127X[104X
    [4X[28XGroup([ (), (), (), (1,2,3), (1,2) ])[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YFinite balls[133X[101X
  
  [33X[0;0YThe  automorphism  groups  of  the  finite labelled balls [23XB_{d,k}[123X lie at the
  center   of   this  package.  The  method  [2XAutBall[102X  ([14X2.2-1[114X)  produces  these
  automorphism groups as iterated wreath products. The result is a permutation
  group on the set of leaves of [23XB_{d,k}[123X.[133X
  
  [1X2.2-1 AutBall[101X
  
  [33X[1;0Y[29X[2XAutBall[102X( [3Xd[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe  local  action [23X\mathrm{Aut}(B_{d,k})[123X as a permutation group of
            the [23Xd\cdot (d-1)^{k-1}[123X leaves of [23XB_{d,k}[123X.[133X
  
  [33X[0;0YThe  arguments  of  this  method  are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X and a
  radius [3Xk[103X [23X\in\mathbb{N}_{0}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=AutBall(3,2);[127X[104X
    [4X[28XGroup([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XSize(G);[127X[104X
    [4X[28X48[128X[104X
  [4X[32X[104X
  
  
  [1X2.3 [33X[0;0YAddresses and leaves[133X[101X
  
  [33X[0;0YThe  vertices  at  distance  [23Xn[123X from the center [23Xb[123X of [23XB_{d,k}[123X are addressed as
  elements of the set[133X
  
  
  [24X[33X[0;6Y\Omega^{(n)}:=\{(\omega_{1},\ldots,\omega_{n})\in\Omega^{n}\mid      \forall
  l\in\{1,\ldots,n-1\}:\ \omega_{l}\neq\omega_{l+1}\},[133X
  
  [124X
  
  [33X[0;0Yi.e.  as  lists  of  length  [23Xn[123X  of  elements  from  [10X[1..d][110X  such that no two
  consecutive   entries   are   equal.  They  are  ordered  according  to  the
  lexicographic order on [23X\Omega^{(n)}[123X. The center [23Xb[123X itself is addressed by the
  empty  list  [10X[][110X.  Note  that the leaves of [23XB_{d,k}[123X correspond to elements of
  [23X\Omega^{(k)}[123X.[133X
  
  [1X2.3-1 BallAddresses[101X
  
  [33X[1;0Y[29X[2XBallAddresses[102X( [3Xd[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya  list of all addresses of vertices in [23XB_{d,k}[123X in ascending order
            with  respect  to  length,  lexicographically  ordered within each
            level. See [2XAddressOfLeaf[102X ([14X2.3-3[114X) and [2XLeafOfAddress[102X ([14X2.3-4[114X) for the
            correspondence  between  the  leaves  of  [23XB_{d,k}[123X and addresses of
            length [3Xk[103X.[133X
  
  [33X[0;0YThe  arguments  of  this  method  are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X and a
  radius [3Xk[103X [23X\in\mathbb{N}_{0}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBallAddresses(3,1);[127X[104X
    [4X[28X[ [  ], [ 1 ], [ 2 ], [ 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XBallAddresses(3,2);[127X[104X
    [4X[28X[ [  ], [ 1 ], [ 2 ], [ 3 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ], [128X[104X
    [4X[28X  [ 3, 1 ], [ 3, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X2.3-2 LeafAddresses[101X
  
  [33X[1;0Y[29X[2XLeafAddresses[102X( [3Xd[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya  list  of  addresses  of  the leaves of [23XB_{d,k}[123X in lexicographic
            order.[133X
  
  [33X[0;0YThe  arguments  of  this  method  are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X and a
  radius [3Xk[103X [23X\in\mathbb{N}_{0}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLeafAddresses(3,2);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X2.3-3 AddressOfLeaf[101X
  
  [33X[1;0Y[29X[2XAddressOfLeaf[102X( [3Xd[103X, [3Xk[103X, [3Xlf[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe  address  of  the  leaf  [3Xlf[103X  of  [23XB_{d,k}[123X  with  respect to the
            lexicographic order.[133X
  
  [33X[0;0YThe  arguments of this method are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X, a radius
  [3Xk[103X [23X\in\mathbb{N}[123X, and a leaf [3Xlf[103X of [23XB_{d,k}[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAddressOfLeaf(3,2,1);[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27XAddressOfLeaf(3,3,1);[127X[104X
    [4X[28X[ 1, 2, 1 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-4 LeafOfAddress[101X
  
  [33X[1;0Y[29X[2XLeafOfAddress[102X( [3Xd[103X, [3Xk[103X, [3Xaddr[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe smallest leaf (integer) whose address has [3Xaddr[103X as a prefix.[133X
  
  [33X[0;0YThe  arguments of this method are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X, a radius
  [3Xk[103X [23X\in\mathbb{N}[123X, and an address [3Xaddr[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLeafOfAddress(3,2,[1,2]);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XLeafOfAddress(3,2,[3]);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XLeafOfAddress(3,2,[]);[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X2.3-5 ImageAddress[101X
  
  [33X[1;0Y[29X[2XImageAddress[102X( [3Xd[103X, [3Xk[103X, [3Xaut[103X, [3Xaddr[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe  address of the image of the vertex represented by the address
            [3Xaddr[103X under the automorphism [3Xaut[103X of [23XB_{d,k}[123X.[133X
  
  [33X[0;0YThe  arguments of this method are a degree [3Xd[103X [23X\in\mathbb{N}_{\ge 3}[123X, a radius
  [3Xk[103X [23X\in\mathbb{N}[123X, an automorphism [3Xaut[103X of [23XB_{d,k}[123X, and an address [3Xaddr[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XImageAddress(3,2,(1,2),[1,2]);[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27XImageAddress(3,2,(1,2),[1]);[127X[104X
    [4X[28X[ 1 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-6 ComposeAddresses[101X
  
  [33X[1;0Y[29X[2XComposeAddresses[102X( [3Xaddr1[103X, [3Xaddr2[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ythe  concatenation of the addresses [3Xaddr1[103X and [3Xaddr2[103X with reduction
            as per [Tor20, Section 3.2].[133X
  
  [33X[0;0YThe arguments of this method are two addresses [3Xaddr1[103X and [3Xaddr2[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XComposeAddresses([1,3],[2,1]);  [127X[104X
    [4X[28X[ 1, 3, 2, 1 ][128X[104X
    [4X[25Xgap>[125X [27XComposeAddresses([1,3,2],[2,1]);[127X[104X
    [4X[28X[ 1, 3, 1 ][128X[104X
  [4X[32X[104X
  
