<p>This chapter contains functions that are related to the discreteness property (D) presented in Proposition 3.12 of <a href="chapBib.html#biBTor20">[Tor20]</a>.</p>
<p>Said proposition shows that for a given <span class="Math">F\le \mathrm{Aut}(B_{d,k})</span> the group <span class="Math">\mathrm{U}_{k}(F)</span> is discrete if and only if the maximal compatible subgroup <span class="Math">C(F)</span> of <span class="Math">F</span> satisfies condition (D):</p>
<p>where <span class="Math">T_{\omega}</span> is the <span class="Math">k-1</span>-neighbourhood of the edge <span class="Math">(b,b_{\omega})</span> inside <span class="Math">B_{d,k}</span>. In other words, <span class="Math">F</span> satisfies (D) if and only if the compatibility set <span class="Math">C_{F}(\mathrm{id},\omega)=\{\mathrm{id}\}</span>. We distinguish between <spanclass="Math">F</span> satisfying condition (D) and <span class="Math">\mathrm{U}_{k}(F)</span> being discrete with the methods <code class="func">SatisfiesD</code> (<a href="chap5.html#X87A11A3E7BDC0549"><span class="RefLink">5.2-1</span></a>) and <code class="func">YieldsDiscreteUniversalGroup</code> (<a href="chap5.html#X7C1C3A8386A91E6C"><span class="RefLink">5.2-2</span></a>) below.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">\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="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>
<p>Subgroups <span class="Math">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="Math">z:F\times\{1,\ldots,d\}\to F</span> that satisfies certain properties, see <a href="chapBib.html#biBTor20">[Tor20, Section 3.2.2]</a>. When <span class="Math">F</span> satisfies just (C), it may still admit an involutive compatibility cocycle. In this case, F admits an extension <span class="Math">\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.html#X80ADE0E379590053"><span class="RefLink">5.3-1</span></a>) and <code class="func">AllInvolutiveCompatibilityCocycles</code> (<a href="chap5.html#X83A26CBF87AB1FD9"><span class="RefLink">5.3-2</span></a>) below.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">\times</span><code class="code">[1..d]</code><span class="Math">\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="Math">=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="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>)), which is compatible (see <code class="func">SatisfiesC</code> (<a href="chap3.html#X8302F28A85F1C4FE"><span class="RefLink">3.3-2</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">F</span>.</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>)), which is compatible (see <code class="func">SatisfiesC</code> (<a href="chap3.html#X8302F28A85F1C4FE"><span class="RefLink">3.3-2</span></a>)).</p>
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