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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>

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<h1><strong class="pkg">UGALY</strong></h1>


<h2>Universal Groups Acting LocallY</h2>

<p>
    4.1.3</p>

<p>
    7 July 2023
  </p>

</div>
<p><b>
    Khalil Hannouch




  </b>
<br />Email: <span class="URL"><a href="mailto:khalil.hannouch@newcastle.edu.au">khalil.hannouch@newcastle.edu.au</a></span>
<br />Homepage: <span class="URL"><a href="https://www.newcastle.edu.au/profile/khalil-hannouch">https://www.newcastle.edu.au/profile/khalil-hannouch</a></span>
<br />Address: <br />Khalil Hannouch<br /> The University of Newcastle<br /> School of Information and Physical Sciences<br /> University Drive<br /> 2308 Callaghan NSW<br /> Australia<br />
</p><p><b>
    Stephan Tornier




  </b>
<br />Email: <span class="URL"><a href="mailto:stephan.tornier@newcastle.edu.au">stephan.tornier@newcastle.edu.au</a></span>
<br />Homepage: <span class="URL"><a href="https://www.newcastle.edu.au/profile/stephan-tornier">https://www.newcastle.edu.au/profile/stephan-tornier</a></span>
<br />Address: <br />Stephan Tornier<br /> The University of Newcastle<br /> School of Information and Physical Sciences<br /> University Drive<br /> 2308 Callaghan NSW<br /> Australia<br />
</p>

<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p><strong class="pkg">UGALY</strong> (<strong class="button">U</strong>niversal <strong class="button">G</strong>roups <strong class="button">A</strong>cting <strong class="button">L</strong>ocall<strong class="button">Y</strong>) is a <strong class="button">GAP</strong> package that provides methods to create, analyse and find local actions of generalised universal groups acting on locally finite regular trees, following Burger-Mozes and Tornier.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p><strong class="pkg">UGALY</strong> is free software; you can redistribute it and/or modify it under the terms of the <span class="URL"><a href="https://www.fsf.org/licenses/gpl.html">GNU General Public License</a></span> as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.</p>

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<h3>Acknowledgements</h3>
<p>Section <a href="chap4.html#X7A8A46F4856570BE"><span class="RefLink">4.6</span></a> is due to Tasman Fell. The second author owes thanks to Marc Burger and George Willis for their support and acknowledges contributions from the SNSF Doc.Mobility fellowship 172120 and the ARC Discovery Project DP120100996 to the development of an early version of this codebase. In its present form, the development of <strong class="pkg">UGALY</strong> was made possible by the ARC Laureate Fellowship FL170100032 and the ARC DECRA Fellowship DE210100180. Finally, we owe thanks to Laurent Bartholdi for guiding us through a reviewing process that has resulted in substantial improvements, and to Max Horn for helping with a documentation issue.</p>

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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X78E0A9867DABCE86">1.1 <span class="Heading">Purpose</span></a>
</span>
</div>
</div>
<div class="ContChap"><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>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X855A2B187C52B82A">2.2 <span class="Heading">Finite balls</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X784EBFE9796B960C">2.2-1 AutBall</a></span>
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<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>
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<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>
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<div class="ContChap"><a href="chap3.html#X7F4E157A827198EA">3 <span class="Heading">Compatibility</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81B0CAE97D161B97">3.1 <span class="Heading">The compatibility condition (C)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X82C839D7794DDBCD">3.2 <span class="Heading">Compatible elements</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84AA17C28591AC1F">3.2-1 AreCompatibleBallElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8206962582F9C96A">3.2-2 CompatibleBallElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7ECC9E9982597E12">3.2-3 <span class="Heading">CompatibilitySet</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X837E15117DB5B1DE">3.2-4 AssembleAutomorphism</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7967612B81C5074E">3.3 <span class="Heading">Compatible subgroups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79D1BC54814345D4">3.3-1 MaximalCompatibleSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8302F28A85F1C4FE">3.3-2 SatisfiesC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FF0262182F3B9E8">3.3-3 CompatibleSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CEBD97183C3399E">3.3-4 ConjugacyClassRepsCompatibleSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84F140AC7AD8D9EE">3.3-5 ConjugacyClassRepsCompatibleGroupsWithProjection</a></span>
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<div class="ContChap"><a href="chap4.html#X7A489A5D79DA9E5C">4 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A91225C7FCEBA7D">4.1 <span class="Heading">Discrete groups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78CD1F4D86BF2A1B">4.1-1 <span class="Heading">LocalActionElement</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X864633EB84B6C73E">4.1-2 <span class="Heading">LocalActionGamma</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8366310185354A2D">4.1-3 <span class="Heading">LocalActionDelta</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F731157843279EC">4.2 <span class="Heading">Maximal extensions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8348A9F887317289">4.2-1 <span class="Heading">LocalActionPhi</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X80EDDA597A66421E">4.3 <span class="Heading">Normal subgroups and partitions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8348A9F887317289">4.3-1 <span class="Heading">LocalActionPhi</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X784DFD007FE78DE9">4.4 <span class="Heading">Abelian quotients</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F4CC068860BA22F">4.4-1 SignHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D8D1FC07BC90971">4.4-2 AbelianizationHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83A7A23D875BFAA2">4.4-3 SpheresProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79341499795BF8D9">4.4-4 LocalActionPi</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X87FE512E7DB7346C">4.5 <span class="Heading">Semidirect products</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F425DFC8760388F">4.5-1 <span class="Heading">CompatibleKernels</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83920A2C7CC46AC9">4.5-2 <span class="Heading">LocalActionSigma</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A8A46F4856570BE">4.6 <span class="Heading">PGL₂ over the p-adic numbers</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E80A4A27E8D846C">4.6-1 LocalActionPGL2Qp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87D6A7B679C668A8">4.6-2 LocalActionPSL2Qp</a></span>
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<div class="ContChap"><a href="chap5.html#X7875F15E81CDBD6C">5 <span class="Heading">Discreteness</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7B8BCB2681070C9C">5.1 <span class="Heading">The discreteness condition (D)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7875F15E81CDBD6C">5.2 <span class="Heading">Discreteness</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87A11A3E7BDC0549">5.2-1 SatisfiesD</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C1C3A8386A91E6C">5.2-2 YieldsDiscreteUniversalGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X85A9B66278AF63D9">5.3 <span class="Heading">Cocycles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80ADE0E379590053">5.3-1 InvolutiveCompatibilityCocycle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83A26CBF87AB1FD9">5.3-2 AllInvolutiveCompatibilityCocycles</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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