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<title>GAP (UGALY) - Chapter 5: Discreteness</title>
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<div class="ChapSects"><a href="chap5_mj.html#X7875F15E81CDBD6C">5 <span class="Heading">Discreteness</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7B8BCB2681070C9C">5.1 <span class="Heading">The discreteness condition (D)</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7875F15E81CDBD6C">5.2 <span class="Heading">Discreteness</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87A11A3E7BDC0549">5.2-1 SatisfiesD</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C1C3A8386A91E6C">5.2-2 YieldsDiscreteUniversalGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85A9B66278AF63D9">5.3 <span class="Heading">Cocycles</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X80ADE0E379590053">5.3-1 InvolutiveCompatibilityCocycle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83A26CBF87AB1FD9">5.3-2 AllInvolutiveCompatibilityCocycles</a></span>
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<h3>5 <span class="Heading">Discreteness</span></h3>

<p>This chapter contains functions that are related to the discreteness property (D) presented in Proposition 3.12 of <a href="chapBib_mj.html#biBTor20">[Tor20]</a>.</p>

<p><a id="X7B8BCB2681070C9C" name="X7B8BCB2681070C9C"></a></p>

<h4>5.1 <span class="Heading">The discreteness condition (D)</span></h4>

<p>Said proposition shows that for a given <span class="SimpleMath">\(F\le \mathrm{Aut}(B_{d,k})\)</span> the group <span class="SimpleMath">\(\mathrm{U}_{k}(F)\)</span> is discrete if and only if the maximal compatible subgroup <span class="SimpleMath">\(C(F)\)</span> of <span class="SimpleMath">\(F\)</span> satisfies condition (D):</p>

<p class="center">\[\forall \omega \in \Omega: F_{T_{\omega}}=\{\mathrm{id}\},\]</p>

<p>where <span class="SimpleMath">\(T_{\omega}\)</span> is the <span class="SimpleMath">\(k-1\)</span>-neighbourhood of the edge <span class="SimpleMath">\((b,b_{\omega})\)</span> inside <span class="SimpleMath">\(B_{d,k}\)</span>. In other words, <span class="SimpleMath">\(F\)</span> satisfies (D) if and only if the compatibility set <span class="SimpleMath">\(C_{F}(\mathrm{id},\omega)=\{\mathrm{id}\}\)</span>. We distinguish between <span class="SimpleMath">\(F\)</span> satisfying condition (D) and <span class="SimpleMath">\(\mathrm{U}_{k}(F)\)</span> being discrete with the methods <code class="func">SatisfiesD</code> (<a href="chap5_mj.html#X87A11A3E7BDC0549"><span class="RefLink">5.2-1</span></a>) and <code class="func">YieldsDiscreteUniversalGroup</code> (<a href="chap5_mj.html#X7C1C3A8386A91E6C"><span class="RefLink">5.2-2</span></a>) below.</p>

<p><a id="X7875F15E81CDBD6C" name="X7875F15E81CDBD6C"></a></p>

<h4>5.2 <span class="Heading">Discreteness</span></h4>

<p><a id="X87A11A3E7BDC0549" name="X87A11A3E7BDC0549"></a></p>

<h5>5.2-1 SatisfiesD</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SatisfiesD</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">F</var> satisfies the discreteness condition (D), and <code class="keyw">false</code> otherwise.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="SimpleMath">\(\le\mathrm{Aut}(B_{d,k})\)</span> (see <code class="func">IsLocalAction</code> (<a href="chap2_mj.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>


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

<p><a id="X7C1C3A8386A91E6C" name="X7C1C3A8386A91E6C"></a></p>

<h5>5.2-2 YieldsDiscreteUniversalGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; YieldsDiscreteUniversalGroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <span class="SimpleMath">\(\mathrm{U}_{k}(F)\)</span> is discrete, and <code class="keyw">false</code> otherwise.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="SimpleMath">\(\le\mathrm{Aut}(B_{d,k})\)</span> (see <code class="func">IsLocalAction</code> (<a href="chap2_mj.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>


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


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalAction(3,2,Group((1,2)));</span>
Group([ (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">YieldsDiscreteUniversalGroup(F);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SatisfiesD(F);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=MaximalCompatibleSubgroup(F);</span>
Group(())
<span class="GAPprompt">gap></span> <span class="GAPinput">SatisfiesD(C);</span>
true
</pre></div>

<p><a id="X85A9B66278AF63D9" name="X85A9B66278AF63D9"></a></p>

<h4>5.3 <span class="Heading">Cocycles</span></h4>

<p>Subgroups <span class="SimpleMath">\(F\le\mathrm{Aut}(B_{d,k})\)</span> that satisfy both (C) and (D) admit an involutive compatibility cocycle, i.e. a map <span class="SimpleMath">\(z:F\times\{1,\ldots,d\}\to F\)</span> that satisfies certain properties, see <a href="chapBib_mj.html#biBTor20">[Tor20, Section 3.2.2]</a>. When <span class="SimpleMath">\(F\)</span> satisfies just (C), it may still admit an involutive compatibility cocycle. In this case, F admits an extension <span class="SimpleMath">\(\Gamma_{z}(F)\le\mathrm{Aut}(B_{d,k})\)</span> that satisfies both (C) and (D). Involutive compatibility cocycles can be searched for using <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5_mj.html#X80ADE0E379590053"><span class="RefLink">5.3-1</span></a>) and <code class="func">AllInvolutiveCompatibilityCocycles</code> (<a href="chap5_mj.html#X83A26CBF87AB1FD9"><span class="RefLink">5.3-2</span></a>) below.</p>

<p><a id="X80ADE0E379590053" name="X80ADE0E379590053"></a></p>

<h5>5.3-1 InvolutiveCompatibilityCocycle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; InvolutiveCompatibilityCocycle</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an involutive compatibility cocycle of <var class="Arg">F</var>, which is a mapping <var class="Arg">F</var><span class="SimpleMath">\(\times\)</span><code class="code">[1..d]</code><span class="SimpleMath">\(\to\)</span><var class="Arg">F</var> with certain properties, if it exists, and <code class="keyw">fail</code> otherwise. When <var class="Arg">k</var> <span class="SimpleMath">\(=1\)</span>, the standard cocycle is returned.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="SimpleMath">\(\le\mathrm{Aut}(B_{d,k})\)</span> (see <code class="func">IsLocalAction</code> (<a href="chap2_mj.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)), which is compatible (see <code class="func">SatisfiesC</code> (<a href="chap3_mj.html#X8302F28A85F1C4FE"><span class="RefLink">3.3-2</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalAction(3,1,AlternatingGroup(3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">z:=InvolutiveCompatibilityCocycle(F);;</span>
<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);; dir:=Random(mt,[1..3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a; Image(z,[a,dir]);</span>
(1,2,3)
(1,2,3)
</pre></div>


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

<p><a id="X83A26CBF87AB1FD9" name="X83A26CBF87AB1FD9"></a></p>

<h5>5.3-2 AllInvolutiveCompatibilityCocycles</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllInvolutiveCompatibilityCocycles</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the list of all involutive compatibility cocycles of <span class="SimpleMath">\(F\)</span>.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="SimpleMath">\(\le\mathrm{Aut}(B_{d,k})\)</span> (see <code class="func">IsLocalAction</code> (<a href="chap2_mj.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)), which is compatible (see <code class="func">SatisfiesC</code> (<a href="chap3_mj.html#X8302F28A85F1C4FE"><span class="RefLink">3.3-2</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S3:=LocalAction(3,1,SymmetricGroup(3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(AllInvolutiveCompatibilityCocycles(S3));</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(AllInvolutiveCompatibilityCocycles(LocalActionGamma(3,SymmetricGroup(3))));</span>
1
</pre></div>


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