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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap1.html">[Previous Chapter]</a>    <a href="chap3.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</a></p>
<p><a id="X8749E1888244CC3D" name="X8749E1888244CC3D"></a></p>
<div class="ChapSects"><a href="chap2.html#X8749E1888244CC3D">2 <span class="Heading">Preliminaries</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86AC008984A3489F">2.1 <span class="Heading">Local actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FCF15167D3A44B7">2.1-1 IsLocalAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81135CA77A3C0F4E">2.1-2 LocalAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84D5C421864EB7FD">2.1-3 LocalActionNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8321CC72807D1096">2.1-4 LocalActionDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BF3EE4D794F8276">2.1-5 LocalActionRadius</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E0E11FC802B5210">2.1-6 LocalAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BE35375787753EE">2.1-7 Projection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87A13DDE8321BEF3">2.1-8 ImageOfProjection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X855A2B187C52B82A">2.2 <span class="Heading">Finite balls</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X784EBFE9796B960C">2.2-1 AutBall</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7A77C1B579101B7C">2.3 <span class="Heading">Addresses and leaves</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84DD7F7881A00315">2.3-1 BallAddresses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X868811D87FEB1AA6">2.3-2 LeafAddresses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78379A547ED7A317">2.3-3 AddressOfLeaf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E43E2B87B97A9BE">2.3-4 LeafOfAddress</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X871D6B8783E17AB8">2.3-5 ImageAddress</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X862D09157F1D9D98">2.3-6 ComposeAddresses</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Preliminaries</span></h3>

<p>We recall the following notation from the Introduction which is essential throughout this manual, cf. <a href="chapBib.html#biBTor20">[Tor20]</a>. Let <span class="Math">\Omega</span> be a set of cardinality <span class="Math">d\in\mathbb{N}_{\ge 3}</span> and let <span class="Math">T_{d}=(V,E)</span> denote the <span class="Math">d</span>-regular tree, following the graph theory notation in <a href="chapBib.html#biBSer80">[Ser80]</a>. A <em>labelling</em> <span class="Math">l</span> of <span class="Math">T_{d}</span> is a map <span class="Math">l:E\to\Omega</span> such that for every <span class="Math">x\in V</span> the restriction <span class="Math">l_{x}:E(x)\to\Omega,\ e\mapsto l(e)</span> is a bijection, and <span class="Math">l(e)=l(\overline{e})</span> for all <span class="Math">e\in E</span>. For every <span class="Math">k\in\mathbb{N}</span>, fix a tree <span class="Math">B_{d,k}</span> which is isomorphic to a ball of radius <span class="Math">k</span> around a vertex in <span class="Math">T_{d}</span> and carry over the labelling of <span class="Math">T_{d}</span> to <span class="Math">B_{d,k}</span> via the chosen isomorphism. We denote the center of <span class="Math">B_{d,k}</span> by <span class="Math">b</span>.</p>

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

<p class="pcenter">\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}.</p>

<p><a id="X86AC008984A3489F" name="X86AC008984A3489F"></a></p>

<h4>2.1 <span class="Heading">Local actions</span></h4>

<p>In this package, local actions <span class="Math">F\le\mathrm{Aut}(B_{d,k})</span> are handled as objects of the category <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>) and have several attributes and properties introduced throughout this manual. Most importantly, a local action always stores the degree <span class="Math">d</span> and the radius <span class="Math">k</span> of the ball <span class="Math">B_{d,k}</span> that it acts on.</p>

<p><a id="X7FCF15167D3A44B7" name="X7FCF15167D3A44B7"></a></p>

<h5>2.1-1 IsLocalAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLocalAction</code>( <var class="Arg">F</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <span class="Math">F</span> is an object of the category <code class="keyw">IsLocalAction</code>, and <code class="keyw">false</code> otherwise.</p>

<p>Local actions <span class="Math">F\le\mathrm{Aut}(B_{d,k})</span> are stored together with their degree (see <code class="func">LocalActionDegree</code> (<a href="chap2.html#X8321CC72807D1096"><span class="RefLink">2.1-4</span></a>)), radius (see <code class="func">LocalActionRadius</code> (<a href="chap2.html#X7BF3EE4D794F8276"><span class="RefLink">2.1-5</span></a>)) as well as other attributes and properties in this category. They can be initialised using the creator operation <code class="func">LocalAction</code> (<a href="chap2.html#X81135CA77A3C0F4E"><span class="RefLink">2.1-2</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=WreathProduct(SymmetricGroup(2),SymmetricGroup(3));</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLocalAction(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=AutBall(3,2);</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLocalAction(H);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=LocalAction(3,2,G);</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLocalAction(K);</span>
true
</pre></div>

<p><a id="X81135CA77A3C0F4E" name="X81135CA77A3C0F4E"></a></p>

<h5>2.1-2 LocalAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalAction</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the regular rooted tree group <span class="Math">G</span> as an object of the category <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>), checking that <var class="Arg">F</var> is indeed a subgroup of <span class="Math">\mathrm{Aut}(B_{d,k})</span>.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}_{0}</span> and a group <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=WreathProduct(SymmetricGroup(2),SymmetricGroup(3));</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLocalAction(G);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LocalAction(3,2,G);</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLocalAction(G);</span>
true
</pre></div>

<p><a id="X84D5C421864EB7FD" name="X84D5C421864EB7FD"></a></p>

<h5>2.1-3 LocalActionNC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionNC</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the regular rooted tree group <span class="Math">G</span> as an object of the category <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>), without checking that <var class="Arg">F</var> is indeed a subgroup of <span class="Math">\mathrm{Aut}(B_{d,k})</span>.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}_{0}</span> and a group <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>.</p>

<p><a id="X8321CC72807D1096" name="X8321CC72807D1096"></a></p>

<h5>2.1-4 LocalActionDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionDegree</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the degree <var class="Arg">d</var> of the ball <span class="Math">B_{d,k}</span> that <span class="Math">F</span> is acting on.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> (see <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A4:=LocalAction(4,1,AlternatingGroup(4));</span>
Alt( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalActionPhi(3,A4);</span>
<permutation group with 18 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalActionDegree(F);</span>
4
</pre></div>

<p><a id="X7BF3EE4D794F8276" name="X7BF3EE4D794F8276"></a></p>

<h5>2.1-5 LocalActionRadius</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionRadius</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the radius <var class="Arg">k</var> of the ball <span class="Math">B_{d,k}</span> that <span class="Math">F</span> is acting on.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> (see <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A4:=LocalAction(4,1,AlternatingGroup(4));</span>
Alt( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalActionPhi(3,A4);</span>
<permutation group with 18 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalActionRadius(F);</span>
3
</pre></div>

<p><a id="X7E0E11FC802B5210" name="X7E0E11FC802B5210"></a></p>

<h5>2.1-6 LocalAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalAction</code>( <var class="Arg">r</var>, <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">addr</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the <var class="Arg">r</var>-local action <span class="Math">\sigma_{r}(</span><var class="Arg">aut</var>,<var class="Arg">addr</var><span class="Math">)</span> of the automorphism <var class="Arg">aut</var> of <span class="Math">B_{d,k}</span> at the vertex represented by the address <var class="Arg">addr</var>.</p>

<p>The arguments of this method are a radius <var class="Arg">r</var>, a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, an automorphism <var class="Arg">aut</var> of <span class="Math">B_{d,k}</span>, and an address <var class="Arg">addr</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=(1,3,5)(2,4,6);; a in AutBall(3,2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalAction(2,3,2,a,[]);</span>
(1,3,5)(2,4,6)
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalAction(1,3,2,a,[]);</span>
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalAction(1,3,2,a,[1]);</span>
(1,2)
</pre></div>


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

<p><a id="X7BE35375787753EE" name="X7BE35375787753EE"></a></p>

<h5>2.1-7 Projection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Projection</code>( <var class="Arg">F</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the restriction of the projection map <span class="Math">\mathrm{Aut}(B_{d,k})\to\mathrm{Aut}(B_{d,r})</span> to <var class="Arg">F</var>.</p>

<p>The arguments of this method are a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, and a projection radius <var class="Arg">r</var> <span class="Math">\le</span> <var class="Arg">k</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalActionGamma(4,3,SymmetricGroup(3));</span>
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) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">pr:=Projection(F,2);</span>
<action homomorphism>
<span class="GAPprompt">gap></span> <span class="GAPinput">mt:=RandomSource(IsMersenneTwister,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Random(mt,F);; Image(pr,a);</span>
(1,2)(3,5)(4,6)
</pre></div>

<p><a id="X87A13DDE8321BEF3" name="X87A13DDE8321BEF3"></a></p>

<h5>2.1-8 ImageOfProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageOfProjection</code>( <var class="Arg">F</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the local action <span class="Math">\sigma_{r}(F,b)\le\mathrm{Aut}(B_{d,r})</span>.</p>

<p>The arguments of this method are a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, and a projection radius <var class="Arg">r</var> <span class="Math">\le</span> <var class="Arg">k</var>. This method uses <code class="func">LocalAction</code> (<a href="chap2.html#X7E0E11FC802B5210"><span class="RefLink">2.1-6</span></a>) on generators rather than <code class="func">Projection</code> (<a href="chap2.html#X7BE35375787753EE"><span class="RefLink">2.1-7</span></a>) on the group to compute the image.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AutBall(3,2);</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageOfProjection(AutBall(3,2),1);</span>
Group([ (), (), (), (1,2,3), (1,2) ])
</pre></div>

<p><a id="X855A2B187C52B82A" name="X855A2B187C52B82A"></a></p>

<h4>2.2 <span class="Heading">Finite balls</span></h4>

<p>The automorphism groups of the finite labelled balls <span class="Math">B_{d,k}</span> lie at the center of this package. The method <code class="func">AutBall</code> (<a href="chap2.html#X784EBFE9796B960C"><span class="RefLink">2.2-1</span></a>) produces these automorphism groups as iterated wreath products. The result is a permutation group on the set of leaves of <span class="Math">B_{d,k}</span>.</p>

<p><a id="X784EBFE9796B960C" name="X784EBFE9796B960C"></a></p>

<h5>2.2-1 AutBall</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutBall</code>( <var class="Arg">d</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the local action <span class="Math">\mathrm{Aut}(B_{d,k})</span> as a permutation group of the <span class="Math">d\cdot (d-1)^{k-1}</span> leaves of <span class="Math">B_{d,k}</span>.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span> and a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}_{0}</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AutBall(3,2);</span>
Group([ (1,2), (3,4), (5,6), (1,3,5)(2,4,6), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(G);</span>
48
</pre></div>

<p><a id="X7A77C1B579101B7C" name="X7A77C1B579101B7C"></a></p>

<h4>2.3 <span class="Heading">Addresses and leaves</span></h4>

<p>The vertices at distance <span class="Math">n</span> from the center <span class="Math">b</span> of <span class="Math">B_{d,k}</span> are addressed as elements of the set</p>

<p class="pcenter">\Omega^{(n)}:=\{(\omega_{1},\ldots,\omega_{n})\in\Omega^{n}\mid \forall l\in\{1,\ldots,n-1\}:\ \omega_{l}\neq\omega_{l+1}\},</p>

<p>i.e. as lists of length <span class="Math">n</span> of elements from <code class="code">[1..d]</code> such that no two consecutive entries are equal. They are ordered according to the lexicographic order on <span class="Math">\Omega^{(n)}</span>. The center <span class="Math">b</span> itself is addressed by the empty list <code class="code">[]</code>. Note that the leaves of <span class="Math">B_{d,k}</span> correspond to elements of <span class="Math">\Omega^{(k)}</span>.</p>

<p><a id="X84DD7F7881A00315" name="X84DD7F7881A00315"></a></p>

<h5>2.3-1 BallAddresses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BallAddresses</code>( <var class="Arg">d</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of all addresses of vertices in <span class="Math">B_{d,k}</span> in ascending order with respect to length, lexicographically ordered within each level. See <code class="func">AddressOfLeaf</code> (<a href="chap2.html#X78379A547ED7A317"><span class="RefLink">2.3-3</span></a>) and <code class="func">LeafOfAddress</code> (<a href="chap2.html#X7E43E2B87B97A9BE"><span class="RefLink">2.3-4</span></a>) for the correspondence between the leaves of <span class="Math">B_{d,k}</span> and addresses of length <var class="Arg">k</var>.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span> and a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}_{0}</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BallAddresses(3,1);</span>
[ [  ], [ 1 ], [ 2 ], [ 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BallAddresses(3,2);</span>
[ [  ], [ 1 ], [ 2 ], [ 3 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ], 
  [ 3, 1 ], [ 3, 2 ] ]
</pre></div>

<p><a id="X868811D87FEB1AA6" name="X868811D87FEB1AA6"></a></p>

<h5>2.3-2 LeafAddresses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeafAddresses</code>( <var class="Arg">d</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of addresses of the leaves of <span class="Math">B_{d,k}</span> in lexicographic order.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span> and a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}_{0}</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeafAddresses(3,2);</span>
[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ] ]
</pre></div>

<p><a id="X78379A547ED7A317" name="X78379A547ED7A317"></a></p>

<h5>2.3-3 AddressOfLeaf</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddressOfLeaf</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">lf</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the address of the leaf <var class="Arg">lf</var> of <span class="Math">B_{d,k}</span> with respect to the lexicographic order.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, and a leaf <var class="Arg">lf</var> of <span class="Math">B_{d,k}</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddressOfLeaf(3,2,1);</span>
[ 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AddressOfLeaf(3,3,1);</span>
[ 1, 2, 1 ]
</pre></div>

<p><a id="X7E43E2B87B97A9BE" name="X7E43E2B87B97A9BE"></a></p>

<h5>2.3-4 LeafOfAddress</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeafOfAddress</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">addr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the smallest leaf (integer) whose address has <var class="Arg">addr</var> as a prefix.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, and an address <var class="Arg">addr</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeafOfAddress(3,2,[1,2]);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">LeafOfAddress(3,2,[3]);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">LeafOfAddress(3,2,[]);</span>
1
</pre></div>

<p><a id="X871D6B8783E17AB8" name="X871D6B8783E17AB8"></a></p>

<h5>2.3-5 ImageAddress</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImageAddress</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">addr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the address of the image of the vertex represented by the address <var class="Arg">addr</var> under the automorphism <var class="Arg">aut</var> of <span class="Math">B_{d,k}</span>.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, an automorphism <var class="Arg">aut</var> of <span class="Math">B_{d,k}</span>, and an address <var class="Arg">addr</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageAddress(3,2,(1,2),[1,2]);</span>
[ 1, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageAddress(3,2,(1,2),[1]);</span>
[ 1 ]
</pre></div>

<p><a id="X862D09157F1D9D98" name="X862D09157F1D9D98"></a></p>

<h5>2.3-6 ComposeAddresses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComposeAddresses</code>( <var class="Arg">addr1</var>, <var class="Arg">addr2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the concatenation of the addresses <var class="Arg">addr1</var> and <var class="Arg">addr2</var> with reduction as per <a href="chapBib.html#biBTor20">[Tor20, Section 3.2]</a>.</p>

<p>The arguments of this method are two addresses <var class="Arg">addr1</var> and <var class="Arg">addr2</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ComposeAddresses([1,3],[2,1]);  </span>
[ 1, 3, 2, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ComposeAddresses([1,3,2],[2,1]);</span>
[ 1, 3, 1 ]
</pre></div>


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