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<Chapter Label="Chapter_Preliminaries">
<Heading>Preliminaries</Heading>

We recall the following notation from the Introduction which is essential throughout this manual, cf. <Cite Key="Tor20"/>. Let <Math>\Omega</Math> be a set of cardinality <Math>d\in\mathbb{N}_{\ge 3}</Math> and let <Math>T_{d}=(V,E)</Math> denote the <Math>d</Math>-regular tree, following the graph theory notation in <Cite Key="Ser80"/>. A <E>labelling</E> <Math>l</Math> of <Math>T_{d}</Math> is a map <Math>l:E\to\Omega</Math> such that for every <Math>x\in V</Math> the restriction <Math>l_{x}:E(x)\to\Omega,\ e\mapsto l(e)</Math> is a bijection, and <Math>l(e)=l(\overline{e})</Math> for all <Math>e\in E</Math>. For every <Math>k\in\mathbb{N}</Math>, fix a tree <Math>B_{d,k}</Math> which is isomorphic to a ball of radius <Math>k</Math> around a vertex in <Math>T_{d}</Math> and carry over the labelling of <Math>T_{d}</Math> to <Math>B_{d,k}</Math> via the chosen isomorphism. We denote the center of <Math>B_{d,k}</Math> by <Math>b</Math>.

For every <Math>x\in V</Math> there is a unique, label-respecting isomorphism <Math>l_{x}^{k}:B(x,k)\to B_{d,k}</Math>. We define the <E><Math>k</Math>-local action</E> <Math>\sigma_{k}(g,x)\in\mathrm{Aut}(B_{d,k})</Math> of an automorphism <Math>g\!\in\!\mathrm{Aut}(T_{d})</Math> at a vertex <Math>x\in V</Math> via the map <Display>\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}.</Display>
<Section Label="Chapter_Preliminaries_Section_Local_actions">
<Heading>Local actions</Heading>

In this package, local actions <Math>F\le\mathrm{Aut}(B_{d,k})</Math> are handled as objects of the category <Ref Filt="IsLocalAction" Label="for IsPermGroup"/> and have several attributes and properties introduced throughout this manual. Most importantly, a local action always stores the degree <Math>d</Math> and the radius <Math>k</Math> of the ball <Math>B_{d,k}</Math> that it acts on.
<ManSection>
 <Filt Arg="F" Name="IsLocalAction" Label="for IsPermGroup"/>
<Returns> <K>true</K> if <Math>F</Math> is an object of the category <K>IsLocalAction</K>, and <K>false</K> otherwise.

</Returns>
<Description>
Local actions <Math>F\le\mathrm{Aut}(B_{d,k})</Math> are stored together with their degree (see <Ref Attr="LocalActionDegree" Label="for IsLocalAction"/>), radius (see <Ref Attr="LocalActionRadius" Label="for IsLocalAction"/>) as well as other attributes and properties in this category. They can be initialised using the creator operation <Ref Oper="LocalAction" Label="for IsInt, IsInt, IsPermGroup"/>.

</Description>
</ManSection>


<Example><![CDATA[
gap> G:=WreathProduct(SymmetricGroup(2),SymmetricGroup(3));
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> IsLocalAction(G);
false
gap> H:=AutBall(3,2);
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> IsLocalAction(H);
true
gap> K:=LocalAction(3,2,G);
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> IsLocalAction(K);
true
]]></Example>

<ManSection>
 <Oper Arg="d,k,F" Name="LocalAction" Label="for IsInt, IsInt, IsPermGroup"/>
<Returns> the regular rooted tree group <Math>G</Math> as an object of the category <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>, checking that <A>F</A> is indeed a subgroup of <Math>\mathrm{Aut}(B_{d,k})</Math>.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math>, a radius <A>k</A> <Math>\in\mathbb{N}_{0}</Math> and a group <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math>.

</Description>
</ManSection>


<Example><![CDATA[
gap> G:=WreathProduct(SymmetricGroup(2),SymmetricGroup(3));
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> IsLocalAction(G);
false
gap> G:=LocalAction(3,2,G);
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> IsLocalAction(G);
true
]]></Example>

<ManSection>
 <Oper Arg="d,k,F" Name="LocalActionNC" Label="for IsInt, IsInt, IsPermGroup"/>
<Returns> the regular rooted tree group <Math>G</Math> as an object of the category <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>, without checking that <A>F</A> is indeed a subgroup of <Math>\mathrm{Aut}(B_{d,k})</Math>.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math>, a radius <A>k</A> <Math>\in\mathbb{N}_{0}</Math> and a group <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math>.

</Description>
</ManSection>

<ManSection>
 <Attr Arg="F" Name="LocalActionDegree" Label="for IsLocalAction"/>
<Returns> the degree <A>d</A> of the ball <Math>B_{d,k}</Math> that <Math>F</Math> is acting on.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).

</Description>
</ManSection>


<Example><![CDATA[
gap> A4:=LocalAction(4,1,AlternatingGroup(4));
Alt( [ 1 .. 4 ] )
gap> F:=LocalActionPhi(3,A4);
<permutation group with 18 generators>
gap> LocalActionDegree(F);
4
]]></Example>

<ManSection>
 <Attr Arg="F" Name="LocalActionRadius" Label="for IsLocalAction"/>
<Returns> the radius <A>k</A> of the ball <Math>B_{d,k}</Math> that <Math>F</Math> is acting on.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).

</Description>
</ManSection>


<Example><![CDATA[
gap> A4:=LocalAction(4,1,AlternatingGroup(4));
Alt( [ 1 .. 4 ] )
gap> F:=LocalActionPhi(3,A4);
<permutation group with 18 generators>
gap> LocalActionRadius(F);
3
]]></Example>

<ManSection>
 <Oper Arg="r,d,k,aut,addr" Name="LocalAction" Label="for r, d, k, aut, addr"/>
<Returns> the <A>r</A>-local action <Math>\sigma_{r}(</Math><A>aut</A>,<A>addr</A><Math>)</Math> of the automorphism <A>aut</A> of <Math>B_{d,k}</Math> at the vertex represented by the address <A>addr</A>.

</Returns>
<Description>
The arguments of this method are a radius <A>r</A>, a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math>, a radius <A>k</A> <Math>\in\mathbb{N}</Math>, an automorphism <A>aut</A> of <Math>B_{d,k}</Math>, and an address <A>addr</A>.

</Description>
</ManSection>


<Example><![CDATA[
gap> a:=(1,3,5)(2,4,6);; a in AutBall(3,2);
true
gap> LocalAction(2,3,2,a,[]);
(1,3,5)(2,4,6)
gap> LocalAction(1,3,2,a,[]);
(1,2,3)
gap> LocalAction(1,3,2,a,[1]);
(1,2)
]]></Example>


<Example><![CDATA[
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> b:=Random(mt,AutBall(3,4));
(1,18,11,5,23,14,4,20,10,7,22,16)(2,17,12,6,24,13,3,19,9,8,21,15)
gap> LocalAction(2,3,4,b,[3,1]);
(1,2)(3,6,4,5)
gap> LocalAction(3,3,4,b,[3,1]);
Error, the sum of input argument r=3 and the length of input argument
addr=[ 3, 1 ] must not exceed input argument k=4
]]></Example>

<ManSection>
 <Oper Arg="F,r" Name="Projection" Label="for F, r"/>
<Returns> the restriction of the projection map <Math>\mathrm{Aut}(B_{d,k})\to\mathrm{Aut}(B_{d,r})</Math> to <A>F</A>.

</Returns>
<Description>
The arguments of this method are a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math>, and a projection radius <A>r</A> <Math>\le</Math> <A>k</A>.

</Description>
</ManSection>


<Example><![CDATA[
gap> F:=LocalActionGamma(4,3,SymmetricGroup(3));
Group([ (1,16,19)(2,15,20)(3,13,18)(4,14,17)(5,10,23)(6,9,24)(7,12,22)
 (8,11,21), (1,9)(2,10)(3,12)(4,11)(5,15)(6,16)(7,13)(8,14)(17,21)(18,22)
 (19,24)(20,23) ])
gap> pr:=Projection(F,2);
<action homomorphism>
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> a:=Random(mt,F);; Image(pr,a);
(1,2)(3,5)(4,6)
]]></Example>

<ManSection>
 <Func Arg="F,r" Name="ImageOfProjection" />
<Returns> the local action <Math>\sigma_{r}(F,b)\le\mathrm{Aut}(B_{d,r})</Math>.

</Returns>
<Description>
The arguments of this method are a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math>, and a projection radius <A>r</A> <Math>\le</Math> <A>k</A>. This method uses <Ref Oper="LocalAction" Label="for r, d, k, aut, addr"/> on generators rather than <Ref Oper="Projection" Label="for F, r"/> on the group to compute the image.

</Description>
</ManSection>


<Example><![CDATA[
gap> AutBall(3,2);
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> ImageOfProjection(AutBall(3,2),1);
Group([ (), (), (), (1,2,3), (1,2) ])
]]></Example>

</Section>

<Section Label="Chapter_Preliminaries_Section_Finite_balls">
<Heading>Finite balls</Heading>

The automorphism groups of the finite labelled balls <Math>B_{d,k}</Math> lie at the center of this package. The method <Ref Func="AutBall"/> produces these automorphism groups as iterated wreath products. The result is a permutation group on the set of leaves of <Math>B_{d,k}</Math>.
<ManSection>
 <Func Arg="d,k" Name="AutBall" />
<Returns> the local action <Math>\mathrm{Aut}(B_{d,k})</Math> as a permutation group of the <Math>d\cdot (d-1)^{k-1}</Math> leaves of <Math>B_{d,k}</Math>.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math> and a radius <A>k</A> <Math>\in\mathbb{N}_{0}</Math>.

</Description>
</ManSection>


<Example><![CDATA[
gap> G:=AutBall(3,2);
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
gap> Size(G);
48
]]></Example>

</Section>

<Section Label="Chapter_Preliminaries_Section_Addresses_and_leaves">
<Heading>Addresses and leaves</Heading>

The vertices at distance <Math>n</Math> from the center <Math>b</Math> of <Math>B_{d,k}</Math> are addressed as elements of the set <Display>\Omega^{(n)}:=\{(\omega_{1},\ldots,\omega_{n})\in\Omega^{n}\mid \forall l\in\{1,\ldots,n-1\}:\ \omega_{l}\neq\omega_{l+1}\},</Display> i.e. as lists of length <Math>n</Math> of elements from <C>[1..d]</C> such that no two consecutive entries are equal. They are ordered according to the lexicographic order on <Math>\Omega^{(n)}</Math>. The center <Math>b</Math> itself is addressed by the empty list <C>[]</C>. Note that the leaves of <Math>B_{d,k}</Math> correspond to elements of <Math>\Omega^{(k)}</Math>.
<ManSection>
 <Func Arg="d,k" Name="BallAddresses" />
<Returns> a list of all addresses of vertices in <Math>B_{d,k}</Math> in ascending order with respect to length, lexicographically ordered within each level. See <Ref Func="AddressOfLeaf"/> and <Ref Func="LeafOfAddress"/> for the correspondence between the leaves of <Math>B_{d,k}</Math> and addresses of length <A>k</A>.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math> and a radius <A>k</A> <Math>\in\mathbb{N}_{0}</Math>.

</Description>
</ManSection>


<Example><![CDATA[
gap> BallAddresses(3,1);
[ [ ], [ 1 ], [ 2 ], [ 3 ] ]
gap> BallAddresses(3,2);
[ [ ], [ 1 ], [ 2 ], [ 3 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ],
 [ 3, 1 ], [ 3, 2 ] ]
]]></Example>

<ManSection>
 <Func Arg="d,k" Name="LeafAddresses" />
<Returns> a list of addresses of the leaves of <Math>B_{d,k}</Math> in lexicographic order.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math> and a radius <A>k</A> <Math>\in\mathbb{N}_{0}</Math>.

</Description>
</ManSection>


<Example><![CDATA[
gap> LeafAddresses(3,2);
[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ] ]
]]></Example>

<ManSection>
 <Func Arg="d,k,lf" Name="AddressOfLeaf" />
<Returns> the address of the leaf <A>lf</A> of <Math>B_{d,k}</Math> with respect to the lexicographic order.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math>, a radius <A>k</A> <Math>\in\mathbb{N}</Math>, and a leaf <A>lf</A> of <Math>B_{d,k}</Math>.

</Description>
</ManSection>


<Example><![CDATA[
gap> AddressOfLeaf(3,2,1);
[ 1, 2 ]
gap> AddressOfLeaf(3,3,1);
[ 1, 2, 1 ]
]]></Example>

<ManSection>
 <Func Arg="d,k,addr" Name="LeafOfAddress" />
<Returns> the smallest leaf (integer) whose address has <A>addr</A> as a prefix.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math>, a radius <A>k</A> <Math>\in\mathbb{N}</Math>, and an address <A>addr</A>.

</Description>
</ManSection>


<Example><![CDATA[
gap> LeafOfAddress(3,2,[1,2]);
1
gap> LeafOfAddress(3,2,[3]);
5
gap> LeafOfAddress(3,2,[]);
1
]]></Example>

<ManSection>
 <Func Arg="d,k,aut,addr" Name="ImageAddress" />
<Returns> the address of the image of the vertex represented by the address <A>addr</A> under the automorphism <A>aut</A> of <Math>B_{d,k}</Math>.

</Returns>
<Description>
The arguments of this method are a degree <A>d</A> <Math>\in\mathbb{N}_{\ge 3}</Math>, a radius <A>k</A> <Math>\in\mathbb{N}</Math>, an automorphism <A>aut</A> of <Math>B_{d,k}</Math>, and an address <A>addr</A>.

</Description>
</ManSection>


<Example><![CDATA[
gap> ImageAddress(3,2,(1,2),[1,2]);
[ 1, 3 ]
gap> ImageAddress(3,2,(1,2),[1]);
[ 1 ]
]]></Example>

<ManSection>
 <Func Arg="addr1,addr2" Name="ComposeAddresses" />
<Returns> the concatenation of the addresses <A>addr1</A> and <A>addr2</A> with reduction as per <Cite Key="Tor20" Where="Section 3.2"/>.

</Returns>
<Description>
The arguments of this method are two addresses <A>addr1</A> and <A>addr2</A>.

</Description>
</ManSection>


<Example><![CDATA[
gap> ComposeAddresses([1,3],[2,1]);
[ 1, 3, 2, 1 ]
gap> ComposeAddresses([1,3,2],[2,1]);
[ 1, 3, 1 ]
]]></Example>

</Section>

</Chapter>

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