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<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap4_mj.html#X7A489A5D79DA9E5C">4 Examples</a>
<div class="ContSect"> <a href="chap4_mj.html#X7A91225C7FCEBA7D">4.1 Discrete groups</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X78CD1F4D86BF2A1B">4.1-1 LocalActionElement</a>

 <a href="chap4_mj.html#X864633EB84B6C73E">4.1-2 LocalActionGamma</a>

 <a href="chap4_mj.html#X8366310185354A2D">4.1-3 LocalActionDelta</a>

</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X7F731157843279EC">4.2 Maximal extensions</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X8348A9F887317289">4.2-1 LocalActionPhi</a>

</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X80EDDA597A66421E">4.3 Normal subgroups and partitions</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X8348A9F887317289">4.3-1 LocalActionPhi</a>

</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X784DFD007FE78DE9">4.4 Abelian quotients</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7F4CC068860BA22F">4.4-1 SignHomomorphism</a>
 <a href="chap4_mj.html#X7D8D1FC07BC90971">4.4-2 AbelianizationHomomorphism</a>
 <a href="chap4_mj.html#X83A7A23D875BFAA2">4.4-3 SpheresProduct</a>
 <a href="chap4_mj.html#X79341499795BF8D9">4.4-4 LocalActionPi</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X87FE512E7DB7346C">4.5 Semidirect products</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7F425DFC8760388F">4.5-1 CompatibleKernels</a>

 <a href="chap4_mj.html#X83920A2C7CC46AC9">4.5-2 LocalActionSigma</a>

</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X7A8A46F4856570BE">4.6 PGL₂ over the p-adic numbers</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7E80A4A27E8D846C">4.6-1 LocalActionPGL2Qp</a>
 <a href="chap4_mj.html#X87D6A7B679C668A8">4.6-2 LocalActionPSL2Qp</a>
</div></div>
</div>

<h3>4 Examples</h3>

Several classes of examples of subgroups of \(\mathrm{Aut}(B_{d,k})\) that satisfy (C) and or (D) are constructed in <a href="chapBib_mj.html#biBTor20">[Tor20]</a> and implemented in this section. For a given permutation group \(F\le S_{d}\), there are always the three local actions \(\Gamma(F)\), \(\Delta(F)\) and \(\Phi(F)\) on \(\mathrm{Aut}(B_{d,2})\) that project onto \(F\). For some \(F\), these are all distinct and yield all universal groups that have \(F\) as their \(1\)-local action, see <a href="chapBib_mj.html#biBTor20">[Tor20, Theorem 3.32]</a>. More examples arise in particular when either point stabilizers in \(F\) are not simple, \(F\) preserves a partition, or \(F\) is not perfect. This section also includes functions to provide the \(k\)-local actions of the groups \(\mathrm{PGL}(2,\mathbb{Q}_{p})\) and \(\mathrm{PSL}(2,\mathbb{Q}_{p})\) acting on \(T_{p+1}\).

<a id="X7A91225C7FCEBA7D" name="X7A91225C7FCEBA7D"></a>

<h4>4.1 Discrete groups</h4>

Here, we implement the local actions \(\Gamma(F),\Delta(F)\le\mathrm{Aut}(B_{d,2})\), both of which satisfy both (C) and (D), see <a href="chapBib_mj.html#biBTor20">[Tor20, Section 3.4.1]</a>.

<a id="X78CD1F4D86BF2A1B" name="X78CD1F4D86BF2A1B"></a>

<h5>4.1-1 LocalActionElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionElement</code>( <var class="Arg">d</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionElement</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionElement</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">s</var>, <var class="Arg">addr</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionElement</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">a</var></dt>
<dd>Returns: the automorphism \(\gamma(\)<var class="Arg">a</var>\()=(\)<var class="Arg">a</var>\(,(\)<var class="Arg">a</var>\()_{\omega\in\Omega})\in\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\) and a permutation <var class="Arg">a</var> \(\in S_d\).

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">a</var></dt>
<dd>Returns: the automorphism \(\gamma^{l}(\)<var class="Arg">a</var>\()\in\mathrm{Aut}(B_{d,l})\) all of whose \(1\)-local actions are given by <var class="Arg">a</var>.

The arguments of this method are a radius <var class="Arg">l</var> \(\in\mathbb{N}\), a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\) and a permutation <var class="Arg">a</var> \(\in S_d\).

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">s</var>, <var class="Arg">addr</var></dt>
<dd>Returns: the automorphism of \(B_{d,l}\) whose \(1\)-local actions are given by <var class="Arg">s</var> at vertices whose address has <var class="Arg">addr</var> as a prefix and are trivial elsewhere.

The arguments of this method are a radius <var class="Arg">l</var> \(\in\mathbb{N}\), a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a permutation <var class="Arg">s</var> \(\in S_d\) and an address <var class="Arg">addr</var> of a vertex in \(B_{d,l}\) whose last entry is fixed by <var class="Arg">s</var>.

</dd>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">z</var></dt>
<dd>Returns: the automorphism \(\gamma_{z}(\)<var class="Arg">aut</var>\()=(\)<var class="Arg">aut</var>\(,(\)<var class="Arg">z</var>\((\)<var class="Arg">aut</var>\(,\omega))_{\omega\in\Omega})\in\mathrm{Aut}(B_{d,k+1})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a radius <var class="Arg">k</var> \(\in\mathbb{N}\), an automorphism <var class="Arg">aut</var> of \(B_{d,k}\), and an involutive compatibility cocycle <var class="Arg">z</var> of a subgroup of \(\mathrm{Aut}(B_{d,k})\) that contains <var class="Arg">aut</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5_mj.html#X80ADE0E379590053">5.3-1</a>)).

</dd>
</dl>

<div class="example"><pre>
gap> LocalActionElement(3,(1,2));
(1,3)(2,4)(5,6)
</pre></div>

<div class="example"><pre>
gap> LocalActionElement(2,3,(1,2));
(1,3)(2,4)(5,6)
gap> LocalActionElement(3,3,(1,2));
(1,5)(2,6)(3,8)(4,7)(9,11)(10,12)
</pre></div>

<div class="example"><pre>
gap> LocalActionElement(3,3,(1,2),[1,3]);
(3,4)
gap> LocalActionElement(3,3,(1,2),[]);
(1,5)(2,6)(3,8)(4,7)(9,11)(10,12)
</pre></div>

<div class="example"><pre>
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> z1:=AllInvolutiveCompatibilityCocycles(S3)[1];;
gap> LocalActionElement(3,1,(1,2),z1);
(1,4)(2,3)(5,6)
gap> z3:=AllInvolutiveCompatibilityCocycles(S3)[3];;
gap> LocalActionElement(3,1,(1,2),z3);
(1,3)(2,4)(5,6)
</pre></div>

<a id="X864633EB84B6C73E" name="X864633EB84B6C73E"></a>

<h5>4.1-2 LocalActionGamma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionGamma</code>( <var class="Arg">d</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionGamma</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionGamma</code>( <var class="Arg">F</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the local action \(\Gamma(\)<var class="Arg">F</var>\()=\{(a,(a)_{\omega})\mid a\in F\}\le\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), and a subgroup <var class="Arg">F</var> of \(S_{d}\).

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the group \(\Gamma^{l}(\)<var class="Arg">F</var>\()\le\mathrm{Aut}(B_{d,l})\).

The arguments of this method are a radius <var class="Arg">l</var> \(\in\mathbb{N}\), a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), and a subgroup <var class="Arg">F</var> of \(S_d\).

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">z</var></dt>
<dd>Returns: the group \(\Gamma_{z}(\)<var class="Arg">F</var>\()=\{(a,(\)<var class="Arg">z</var>\((a,\omega))_{\omega\in\Omega})\mid a\in\)<var class="Arg">F</var>\(\}\le\mathrm{Aut}(B_{d,k+1})\).

The arguments of this method are a local action <var class="Arg">F</var> \(\le\mathrm{Aut}(B_{d,k})\) and an involutive compatibility cocycle <var class="Arg">z</var> of <var class="Arg">F</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5_mj.html#X80ADE0E379590053">5.3-1</a>)).

</dd>
</dl>

<div class="example"><pre>
gap> F:=TransitiveGroup(4,3);;
gap> LocalActionGamma(4,F);
Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,8)(2,7)(3,9)(4,5)(10,12) ])
</pre></div>

<div class="example"><pre>
gap> LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> LocalActionGamma(2,3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> LocalActionGamma(3,3,SymmetricGroup(3));
Group([ (1,8,10)(2,7,9)(3,5,12)(4,6,11), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12) ])
</pre></div>

<div class="example"><pre>
gap> F:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(F);;
gap> H:=LocalActionPi(2,3,F,rho,[1]);;
gap> z:=InvolutiveCompatibilityCocycle(H);;
gap> g:=LocalActionGamma(H,z);;
gap> [NrMovedPoints(g),TransitiveIdentification(g)];
[ 12, 8 ]
</pre></div>

<a id="X8366310185354A2D" name="X8366310185354A2D"></a>

<h5>4.1-3 LocalActionDelta</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionDelta</code>( <var class="Arg">d</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionDelta</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the group \(\Delta(\)<var class="Arg">F</var>\()\le\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), and a transitive subgroup <var class="Arg">F</var> of \(S_{d}\).

</dd>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">C</var></dt>
<dd>Returns: the group \(\Delta(\)<var class="Arg">F</var>\(,\)<var class="Arg">C</var>\()\le\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a transitive subgroup <var class="Arg">F</var> of \(S_d\), and a central subgroup <var class="Arg">C</var> of the stabilizer <var class="Arg">F</var>\(_{1}\) of \(1\) in <var class="Arg">F</var>.

</dd>
</dl>

<div class="example"><pre>
gap> F:=SymmetricGroup(3);;
gap> D:=LocalActionDelta(3,F);
Group([ (1,3,6)(2,4,5), (1,3)(2,4), (1,2)(3,4)(5,6) ])
gap> F1:=Stabilizer(F,1);;
gap> D1:=LocalActionDelta(3,F,F1);
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6), (1,2)(3,4)(5,6) ])
gap> D=D1;
false
gap> G:=AutBall(3,2);;
gap> D^G=D1^G;
true
</pre></div>

<div class="example"><pre>
gap> F:=PrimitiveGroup(5,3);
AGL(1, 5)
gap> F1:=Stabilizer(F,1);
Group([ (2,3,4,5) ])
gap> C:=Group((2,4)(3,5));
Group([ (2,4)(3,5) ])
gap> Index(F1,C);
2
gap> Index(LocalActionDelta(5,F,F1),LocalActionDelta(5,F,C));
2
</pre></div>

<a id="X7F731157843279EC" name="X7F731157843279EC"></a>

<h4>4.2 Maximal extensions</h4>

For any \(F\le\mathrm{Aut}(B_{d,k})\) that satisfies (C), the group \(\Phi(F)\le\mathrm{Aut}(B_{d,k+1})\) is the maximal extension of \(F\) that satisfies (C) as well. It stems from the action of \(\mathrm{U}_{k}(F)\) on balls of radius \(k+1\) in \(T_{d}\).

<a id="X8348A9F887317289" name="X8348A9F887317289"></a>

<h5>4.2-1 LocalActionPhi</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">l</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the argument <var class="Arg">F</var></dt>
<dd>Returns: the group \(\Phi_{k}(\)<var class="Arg">F</var>\()=\{(a,(a_{\omega})_{\omega})\mid a\in \)<var class="Arg">F</var>\(,\ \forall \omega\in\Omega:\ a_{\omega}\in C_{F}(a,\omega)\}\le\mathrm{Aut}(B_{d,k+1})\).

The argument of this method is a local action <var class="Arg">F</var> \(\le\mathrm{Aut}(B_{d,k})\).

</dd>
<dt>for the arguments <var class="Arg">l</var>, <var class="Arg">F</var></dt>
<dd>Returns: the group \(\Phi^{l}(\)<var class="Arg">F</var>\()=\Phi_{l-1}\circ\cdots\circ\Phi_{k+1}\circ\Phi_{k}(\)<var class="Arg">F</var>\()\le\mathrm{Aut}(B_{d,l})\).

The arguments of this method are a radius <var class="Arg">l</var> \(\in\mathbb{N}\) and a local action <var class="Arg">F</var> \(\le\mathrm{Aut}(B_{d,k})\).

</dd>
</dl>

<div class="example"><pre>
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> LocalActionPhi(S3);
Group([ (), (1,4,5)(2,3,6), (1,3)(2,4)(5,6), (1,2), (3,4), (5,6) ])
gap> last=AutBall(3,2);
true
gap> A3:=LocalAction(3,1,AlternatingGroup(3));;
gap> LocalActionPhi(A3);
Group([ (), (1,4,5)(2,3,6) ])
gap> last=LocalActionGamma(3,AlternatingGroup(3));
true
</pre></div>

<div class="example"><pre>
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> groups:=ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3);
[ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]),
 Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]),
 Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]),
 Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]),
 Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ]
gap> for G in groups do Print(Size(G),",",Size(LocalActionPhi(G)),"\n"); od;
6,6
12,12
24,192
24,192
48,3072
</pre></div>

<div class="example"><pre>
gap> LocalActionPhi(3,LocalAction(4,1,SymmetricGroup(4)));
<permutation group with 34 generators>
gap> last=AutBall(4,3);
true
</pre></div>

<div class="example"><pre>
gap> rho:=SignHomomorphism(SymmetricGroup(3));;
gap> F:=LocalActionPi(2,3,SymmetricGroup(3),rho,[1]);; Size(F);
24
gap> P:=LocalActionPhi(4,F);; Size(P);
12288
gap> IsSubgroup(AutBall(3,4),P);
true
gap> SatisfiesC(P);
true
</pre></div>

<a id="X80EDDA597A66421E" name="X80EDDA597A66421E"></a>

<h4>4.3 Normal subgroups and partitions</h4>

When point stabilizers in \(F\le S_{d}\) are not simple, or \(F\) preserves a partition, more universal groups can be constructed as follows.

<a id="X8348A9F887317289" name="X8348A9F887317289"></a>

<h5>4.3-1 LocalActionPhi</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPhi</code>( <var class="Arg">F</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">N</var></dt>
<dd>Returns: the group \(\Phi(\)<var class="Arg">F</var>\(,\)<var class="Arg">N</var>\()\le\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a transitive permutation group <var class="Arg">F</var> \(\le S_{d}\) and a normal subgroup <var class="Arg">N</var> of the stabilizer <var class="Arg">F</var>\(_{1}\) of \(1\) in <var class="Arg">F</var>.

</dd>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">P</var></dt>
<dd>Returns: the group \(\Phi(\)<var class="Arg">F</var>\(,\)<var class="Arg">P</var>\()=\{(a,(a_{\omega})_{\omega})\mid a\in \)<var class="Arg">F</var>\(,\ a_{\omega}\in C_{F}(a,\omega)\) constant w.r.t. <var class="Arg">P</var>\(\}\le\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\) and a permutation group <var class="Arg">F</var> \(\le S_{d}\) and a partition <var class="Arg">P</var> of <code class="code">[1..d]</code> preserved by <var class="Arg">F</var>.

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">P</var></dt>
<dd>Returns: the group \(\Phi_{k}(\)<var class="Arg">F</var>\(,\)<var class="Arg">P</var>\()=\{(\alpha,(\alpha_{\omega})_{\omega})\mid \alpha\in \textit{F},\ \alpha_{\omega}\in C_{F}(\alpha,\omega)\) constant w.r.t. <var class="Arg">P</var>\(\}\le\mathrm{Aut}(B_{d,k+1})\).

The arguments of this method are a local action <var class="Arg">F</var> \(\le\mathrm{Aut}(B_{d,k})\) and a partition <var class="Arg">P</var> of <code class="code">[1..d]</code> preserverd by \(\pi\)<var class="Arg">F</var> \(\le S_{d}\). This method assumes that all compatibility sets with respect to a partition element are non-empty and that all compatibility sets of the identity with respect to a partition element are non-trivial.

</dd>
</dl>

<div class="example"><pre>
gap> F:=SymmetricGroup(4);;
gap> F1:=Stabilizer(F,1);
Sym( [ 2 .. 4 ] )
gap> grps:=NormalSubgroups(F1);
[ Sym( [ 2 .. 4 ] ), Alt( [ 2 .. 4 ] ), Group(()) ]
gap> N:=grps[2];
Alt( [ 2 .. 4 ] )
gap> LocalActionPhi(4,F,N);
Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,4)(2,5)(3,6)(7,8)(10,11), (1,2,3) ])
gap> Index(F1,N);
2
gap> Index(LocalActionPhi(4,F,F1),LocalActionPhi(4,F,N));
16
</pre></div>

<div class="example"><pre>
gap> F:=TransitiveGroup(4,3);
D(4)
gap> P:=Blocks(F,[1..4]);
[ [ 1, 3 ], [ 2, 4 ] ]
gap> G:=LocalActionPhi(4,F,P);
Group([ (1,5,9,10)(2,6,7,11)(3,4,8,12), (1,8)(2,7)(3,9)(4,5)(10,12), (1,3)
 (8,9), (4,5)(10,12) ])
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> aut:=Random(mt,G);
(1,3)(4,12)(5,10)(6,11)(8,9)
gap> LocalAction(1,4,2,aut,[1]); LocalAction(1,4,2,aut,[3]);
(2,4)
(2,4)
gap> LocalAction(1,4,2,aut,[2]); LocalAction(1,4,2,aut,[4]);
(1,3)(2,4)
(1,3)(2,4)
</pre></div>

<div class="example"><pre>
gap> H:=TransitiveGroup(4,3);
D(4)
gap> P:=Blocks(H,[1..4]);
[ [ 1, 3 ], [ 2, 4 ] ]
gap> F:=LocalActionPhi(4,H,P);;
gap> G:=LocalActionPhi(F,P);;
gap> SatisfiesC(G);
true
</pre></div>

<a id="X784DFD007FE78DE9" name="X784DFD007FE78DE9"></a>

<h4>4.4 Abelian quotients</h4>

When a permutation group \(F\le S_{d}\) is not perfect, i.e. it admits an abelian quotient \(\rho:F\twoheadrightarrow A\), more universal groups can be constructed by imposing restrictions of the form \(\prod_{r\in R}\prod_{x\in S(b,r)}\rho(\sigma_{1}(\alpha,x))=1\) on elements \(\alpha\in\Phi^{k}(F)\le\mathrm{Aut}(B_{d,k})\).

<a id="X7F4CC068860BA22F" name="X7F4CC068860BA22F"></a>

<h5>4.4-1 SignHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignHomomorphism</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the sign homomorphism from <var class="Arg">F</var> to \(S_{2}\).

The argument of this method is a permutation group <var class="Arg">F</var> \(\le S_{d}\). This method can be used as an example for the argument <var class="Arg">rho</var> in the methods <code class="func">SpheresProduct</code> (<a href="chap4_mj.html#X83A7A23D875BFAA2">4.4-3</a>) and <code class="func">LocalActionPi</code> (<a href="chap4_mj.html#X79341499795BF8D9">4.4-4</a>).

<div class="example"><pre>
gap> F:=SymmetricGroup(3);;
gap> sign:=SignHomomorphism(F);
MappingByFunction( Sym( [ 1 .. 3 ] ), Sym( [ 1 .. 2 ] ), function( g ) ... end )
gap> Image(sign,(2,3));
(1,2)
gap> Image(sign,(1,2,3));
()
</pre></div>

<a id="X7D8D1FC07BC90971" name="X7D8D1FC07BC90971"></a>

<h5>4.4-2 AbelianizationHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianizationHomomorphism</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the homomorphism from <var class="Arg">F</var> to \(F/[F,F]\).

The argument of this method is a permutation group <var class="Arg">F</var> \(\le S_{d}\). This method can be used as an example for the argument <var class="Arg">rho</var> in the methods <code class="func">SpheresProduct</code> (<a href="chap4_mj.html#X83A7A23D875BFAA2">4.4-3</a>) and <code class="func">LocalActionPi</code> (<a href="chap4_mj.html#X79341499795BF8D9">4.4-4</a>).

<div class="example"><pre>
gap> F:=PrimitiveGroup(5,3);
AGL(1, 5)
gap> ab:=AbelianizationHomomorphism(PrimitiveGroup(5,3));
[ (2,3,4,5), (1,2,3,5,4) ] -> [ f1, <identity> of ... ]
gap> Elements(Range(ab));
[ <identity> of ..., f1, f2, f1*f2 ]
gap> StructureDescription(Range(ab));
"C4"
</pre></div>

<a id="X83A7A23D875BFAA2" name="X83A7A23D875BFAA2"></a>

<h5>4.4-3 SpheresProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpheresProduct</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut</var>, <var class="Arg">rho</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the product \(\prod_{r\in R}\prod_{x\in S(b,r)}\)<var class="Arg">rho</var>\((\sigma_{1}(\)<var class="Arg">aut</var>\(,x))\in\mathrm{im}(\)<var class="Arg">rho</var>\()\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a radius <var class="Arg">k</var> \(\in\mathbb{N}\), an automorphism <var class="Arg">aut</var> of \(B_{d,k}\) all of whose \(1\)-local actions are in the domain of the homomorphism <var class="Arg">rho</var> from a subgroup of \(S_d\) to an abelian group, and a sublist <var class="Arg">R</var> of <code class="code">[0..k-1]</code>. This method is used in the implementation of <code class="func">LocalActionPi</code> (<a href="chap4_mj.html#X79341499795BF8D9">4.4-4</a>).

<div class="example"><pre>
gap> rho:=SignHomomorphism(SymmetricGroup(3));;
gap> SpheresProduct(3,2,LocalActionElement(2,3,(1,2)),rho,[0]);
(1,2)
gap> SpheresProduct(3,2,LocalActionElement(2,3,(1,2)),rho,[0,1]);
()
</pre></div>

<div class="example"><pre>
gap> F:=PrimitiveGroup(5,3);
AGL(1, 5)
gap> rho:=AbelianizationHomomorphism(F);;
gap> Elements(Range(rho));
[ <identity> of ..., f1, f2, f1*f2 ]
gap> StructureDescription(Range(rho));
"C4"
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> aut:=Random(mt,F);
(1,4,3,5)
gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[2]);
<identity> of ...
gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[1,2]);
f1
gap> SpheresProduct(5,3,LocalActionElement(3,5,aut),rho,[0,1,2]);
f2
</pre></div>

<a id="X79341499795BF8D9" name="X79341499795BF8D9"></a>

<h5>4.4-4 LocalActionPi</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPi</code>( <var class="Arg">l</var>, <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">rho</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the group \(\Pi^{l}(\)<var class="Arg">F</var>\(,\)<var class="Arg">rho</var>\(,\)<var class="Arg">R</var>\()=\{\alpha\in\Phi^{l}(F)\mid \prod_{r\in R}\prod_{x\in S(b,r)}\)<var class="Arg">rho</var>\((\sigma_{1}(\alpha,x))=1\}\le\mathrm{Aut}(B_{d,l})\).

The arguments of this method are a degree <var class="Arg">l</var> \(\in\mathbb{N}_{\ge 2}\), a radius <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a permutation group <var class="Arg">F</var> \(\le S_d\), a homomorphism \(\rho\) from <var class="Arg">F</var> to an abelian group that is surjective on every point stabilizer in <var class="Arg">F</var>, and a non-empty, non-zero subset <var class="Arg">R</var> of <code class="code">[0..l-1]</code> that contains \(l-1\).

<div class="example"><pre>
gap> F:=LocalAction(5,1,PrimitiveGroup(5,3));
AGL(1, 5)
gap> rho1:=AbelianizationHomomorphism(F);;
gap> rho2:=SignHomomorphism(F);;
gap> LocalActionPi(3,5,F,rho1,[0,1,2]);
<permutation group with 4 generators>
gap> Index(LocalActionPhi(3,F),last);
4
gap> LocalActionPi(3,5,F,rho2,[0,1,2]);
<permutation group with 5 generators>
gap> Index(LocalActionPhi(3,F),last);
2
</pre></div>

<a id="X87FE512E7DB7346C" name="X87FE512E7DB7346C"></a>

<h4>4.5 Semidirect products</h4>

When a subgroup \(F\le\mathrm{Aut}(B_{d,k})\) satisfies (C) and admits an involutive compatibility cocycle \(z\) (which is automatic when \(k=1\)) one can characterise the kernels \(K\le\Phi_{k}(F)\cap\ker(\pi_{k})\) that fit into a \(z\)-split exact sequence \(1\to K\to\Sigma(F,K)\to F\to 1\) for some subgroup \(\Sigma(F,K)\le\mathrm{Aut}(B_{d,k+1})\) that satisfies (C). This characterisation is implemented in this section.

<a id="X7F425DFC8760388F" name="X7F425DFC8760388F"></a>

<h5>4.5-1 CompatibleKernels</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleKernels</code>( <var class="Arg">d</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleKernels</code>( <var class="Arg">F</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var></dt>
<dd>Returns: the list of kernels \(K\le\prod_{\omega\in\Omega}F_{\omega}\cong\ker\pi\le\mathrm{Aut}(B_{d,2})\) that are preserved by the action <var class="Arg">F</var> \(\curvearrowright\prod_{\omega\in\Omega}F_{\omega}\), \(a\cdot(a_{\omega})_{\omega}:=(aa_{a^{-1}\omega}a^{-1})_{\omega}\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), and a permutation group <var class="Arg">F</var> \(\le S_{d}\). The kernels output by this method are compatible with <var class="Arg">F</var> with respect to the standard cocycle (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5_mj.html#X80ADE0E379590053">5.3-1</a>)) and can be used in the method <code class="func">LocalActionSigma</code> (<a href="chap4_mj.html#X83920A2C7CC46AC9">4.5-2</a>).

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">z</var></dt>
<dd>Returns: the list of kernels \(K\le\Phi_{k}(F)\cap\ker(\pi_{k})\le\mathrm{Aut}(B_{d,k+1})\) that are normalized by \(\Gamma_{z}(\)<var class="Arg">F</var>\()\) and such that for all \(k\in K\) and \(\omega\in\Omega\) there is \(k_{\omega}\in K\) with \(\mathrm{pr}_{\omega}k_{\omega}=z(\mathrm{pr}_{\omega}k,\omega)^{-1}\).

The arguments of this method are a local action <var class="Arg">F</var> \(\le\mathrm{Aut}(B_{d,k})\) that satisfies (C) and an involutive compatibility cocycle <var class="Arg">z</var> of <var class="Arg">F</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5_mj.html#X80ADE0E379590053">5.3-1</a>)). It can be used in the method <code class="func">LocalActionSigma</code> (<a href="chap4_mj.html#X83920A2C7CC46AC9">4.5-2</a>).

</dd>
</dl>

<div class="example"><pre>
gap> CompatibleKernels(3,SymmetricGroup(3));
[ Group(()), Group([ (1,2)(3,4)(5,6) ]), Group([ (3,4)(5,6), (1,2)(5,6) ]),
 Group([ (5,6), (3,4), (1,2) ]) ]
</pre></div>

<div class="example"><pre>
gap> P:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(P);;
gap> F:=LocalActionPi(2,3,P,rho,[1]);;
gap> z:=InvolutiveCompatibilityCocycle(F);;
gap> CompatibleKernels(F,z);
[ Group(()), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) ]),
 Group([ (1,2)(3,4)(5,6)(7,8), (5,6)(7,8)(9,10)(11,12) ]),
 Group([ (5,6)(7,8), (1,2)(3,4), (9,10)(11,12) ]) ]
</pre></div>

<a id="X83920A2C7CC46AC9" name="X83920A2C7CC46AC9"></a>

<h5>4.5-2 LocalActionSigma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionSigma</code>( <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">K</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionSigma</code>( <var class="Arg">F</var>, <var class="Arg">K</var>, <var class="Arg">z</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt>for the arguments <var class="Arg">d</var>, <var class="Arg">F</var>, <var class="Arg">K</var></dt>
<dd>Returns: the semidirect product \(\Sigma(\)<var class="Arg">F</var>\(,\)<var class="Arg">K</var>\()\le\mathrm{Aut}(B_{d,2})\).

The arguments of this method are a degree <var class="Arg">d</var> \(\in\mathbb{N}_{\ge 3}\), a subgroup <var class="Arg">F</var> of \(S_{d}\) and a compatible kernel <var class="Arg">K</var> for <var class="Arg">F</var> (see <code class="func">CompatibleKernels</code> (<a href="chap4_mj.html#X7F425DFC8760388F">4.5-1</a>)).

</dd>
<dt>for the arguments <var class="Arg">F</var>, <var class="Arg">K</var>, <var class="Arg">z</var></dt>
<dd>Returns: the semidirect product \(\Sigma_{z}(\)<var class="Arg">F</var>\(,\)<var class="Arg">K</var>\()\le\mathrm{Aut}(B_{d,k+1})\).

The arguments of this method are a local action <var class="Arg">F</var> of \(\mathrm{Aut}(B_{d,k})\) that satisfies (C) and a kernel <var class="Arg">K</var> that is compatible for <var class="Arg">F</var> with respect to the involutive compatibility cocycle <var class="Arg">z</var> (see <code class="func">InvolutiveCompatibilityCocycle</code> (<a href="chap5_mj.html#X80ADE0E379590053">5.3-1</a>) and <code class="func">CompatibleKernels</code> (<a href="chap4_mj.html#X7F425DFC8760388F">4.5-1</a>)) of <var class="Arg">F</var>.

</dd>
</dl>

<div class="example"><pre>
gap> S3:=SymmetricGroup(3);;
gap> kernels:=CompatibleKernels(3,S3);
[ Group(()), Group([ (1,2)(3,4)(5,6) ]), Group([ (3,4)(5,6), (1,2)(5,6) ]),
 Group([ (5,6), (3,4), (1,2) ]) ]
gap> for K in kernels do Print(Size(LocalActionSigma(3,S3,K)),"\n"); od;
6
12
24
48
</pre></div>

<div class="example"><pre>
gap> P:=SymmetricGroup(3);;
gap> rho:=SignHomomorphism(P);;
gap> F:=LocalActionPi(2,3,P,rho,[1]);;
gap> z:=InvolutiveCompatibilityCocycle(F);;
gap> kernels:=CompatibleKernels(F,z);
[ Group(()), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) ]),
 Group([ (1,2)(3,4)(5,6)(7,8), (5,6)(7,8)(9,10)(11,12) ]),
 Group([ (5,6)(7,8), (1,2)(3,4), (9,10)(11,12) ]) ]
gap> for K in kernels do Print(Size(LocalActionSigma(F,K,z)),"\n"); od;
24
48
96
192
</pre></div>

<a id="X7A8A46F4856570BE" name="X7A8A46F4856570BE"></a>

<h4>4.6 PGL₂ over the p-adic numbers</h4>

Here, we implement functions to provide the \(k\)-local actions of the groups \(\mathrm{PGL}(2,\mathbb{Q}_{p})\) and \(\mathrm{PSL}(2,\mathbb{Q}_{p})\) acting on \(T_{p+1}\). This section is due to Tasman Fell.

<a id="X7E80A4A27E8D846C" name="X7E80A4A27E8D846C"></a>

<h5>4.6-1 LocalActionPGL2Qp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPGL2Qp</code>( <var class="Arg">p</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the subgroup of \(\mathrm{Aut}(B_{p+1,k})\) induced by the action of \(\mathrm{PGL}(2,\mathbb{Z}_{p})\) on the ball of radius <var class="Arg">k</var> around the vertex corresponding to the identity lattice of the Bruhat-Tits tree of \(\mathrm{PGL}(2,\mathbb{Q}_{p})\).

The arguments of this method are a prime <var class="Arg">p</var> and a radius <var class="Arg">k</var> \(\in\mathbb{N}_{\ge 1}\).

<div class="example"><pre>
gap> LocalActionPGL2Qp(3,1)=SymmetricGroup(4);
true
gap> F:=LocalActionPGL2Qp(5,3);; Size(F);
1875000
gap> SatisfiesC(F);
true
</pre></div>

<a id="X87D6A7B679C668A8" name="X87D6A7B679C668A8"></a>

<h5>4.6-2 LocalActionPSL2Qp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LocalActionPSL2Qp</code>( <var class="Arg">p</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the subgroup of \(\mathrm{Aut}(B_{p+1,k})\) induced by the action of \(\mathrm{PSL}(2,\mathbb{Z}_{p})\) on the ball of radius <var class="Arg">k</var> around the vertex corresponding to the identity lattice of the Bruhat-Tits tree of \(\mathrm{PGL}(2,\mathbb{Q}_{p})\).

The arguments of this method are a prime <var class="Arg">p</var> and a radius <var class="Arg">k</var> \(\in\mathbb{N}_{\ge 1}\).

<div class="example"><pre>
gap> LocalActionPSL2Qp(3,1)=AlternatingGroup(4);
true
gap> F:=LocalActionPSL2Qp(5,3);; Size(F);
937500
gap> SatisfiesC(F);
true
</pre></div>

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