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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="chapInd.html">Ind</a>  </div>

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<h1>walrus</h1>


<h2>A new approach to proving hyperbolicity</h2>

<p>
    0.9991</p>

<p>
    21 February 2022
  </p>

</div>
<p><b>
    Markus Pfeiffer




  </b>
< [,,,7129,132241274281,]"S^2~^1)12]java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
<br />Homepage: <span class="URL"><a href="http://www.morphism.de/~markusp/">http://www.morphism.de/~markusp/</a></span>
<br />Address: <br />School of Computer Science<br /> University of St Andrews<br /> Jack Cole Building, North Haugh<br /> St Andrews, Fife, KY16 9SX<br /> United Kingdom<br />
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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><[[,,91129,40184187,,]"S2~^)12]
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A489A5D79DA9E5C">1.1 <span class="Heading">Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DE9B0FF7EEC5D80">1.1-1 TriangleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7A3CD8E481C3E802">1.1-2 TriangleCommutatorQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X78BB0EB087263795">1.1-3 RandomTriangleQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7968AC7887291888">1.1-4 JackButtonGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X863AAC9078647B94">1.1-5 RandomPregroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DEF4DA6781CCE1B">1.1-6 RandomPregroupWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X861436A7862A92D2">1.2 <span class="Heading">Testing Hyperbolicity</span></a>
</span>
</div>
<lass="ContSect>span class="tocline"><span class="nocss"> </span><a href="chap1.html#X792204DC7B217ECC">1.3 <span class="Heading">The MAGMA-compatible interface</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X84CF717F7CE7F8A4">2 <span class="Heading">Pregroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F44CAE5832B8511">2.1 <span class="Heading">Creating Pregroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E37DF907C05522C">2.1-1 PregroupByTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8137D096798C1BC8">2.1-2 PregroupByRedRelators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8632A365791D87B7">2.1-3 PregroupOfFreeProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8344F038858426A6">2.1-4 PregroupOfFreeGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B1844C180E739B9">2.2 <span class="Heading">Filters and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86AA05DA7B7BA64B">2.2-1 IsPregroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78E417B3823387C2">2.2-2 IsPregroupTableRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84F97B4D79FBAD44">2.2[2832108113,,44,49,,298329]"S",
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X82866B1A7D018BB7">2.2-4 IsPregroupOfFreeProductRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8053940287FA4077">2.3 <span class="Heading">Attributes, Properties, and Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7946D406823A935B">2.3-1 []</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X80EC827C83DB818B">2.3-2 IntermultPairs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C1AE0D983EF9693">2.3-3 One</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8761B5A57CC4DEA8">2.3-4 MultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BF3B344878B252F">2.3-5 SetPregroupElementNames</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X82B7E2227A94AC98">2.3-6 PregroupElementNames</a></span>
</iv></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7E4700087FC3F933">2.4 <span class="Heading">Elements of Pregroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7CACF96A831CEB1F">2.4-1 IsElementOfPregroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X80185F7579C23714">2.4-2 IsElementOfPregroupRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X838ECC967A2377BA">2.4-3 IsElementOfPregroupOfFreeGroupRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X784ACAE2800C3E7E">2.4-4 PregroupOf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84A42E6B7D542AD1">2.4-5 IsDefinedMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86FC5FDF87C3664A">2.4-6 IsIntermultPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87004CD07832F9DB">2.4-7 PregroupInverse</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X870A0A6A7CAE4680">2.5 <span class="Heading">Small Pregroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7AAEA96C79A55FB3">2.5-1 NrSmallPregroups</a></span>
<span class=ContSS<br /<span class="nocss>&bsp <span>="chap2.html#X7FD30E6F812704C8">2.5-2 SmallPregroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X7A05461687E1FA84">3 <span class="Heading">Pregroup Presentations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7AFDBDC078777B90">3.1 <span class="Heading">Concepts</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83BEFE747F7726F5">3.1-1 <span class="Heading">Locations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87FA460D81566861">3.1- [[28,3128133,,57,289,98,303,L(1)]
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C701DBF7BAE649A">3.2 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81378EEB81DF801A">3.2-1 IsPregroupLocation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85EC64DC8116AF77">3.2-2 InLetter</a></span>
<span class="ContSS"><br /><span <[[2835119126,,82,]"S^2~S^1"]
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81E1F7F984EC37C2">3.2-4 Places</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C88C1227C073A36">3.2-5 NextLocation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83B0E7F778108E23">3.2-6 PrevLocation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X796A41AD85E9AD40">3.2-7 __ID</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85C45E1578DB9E13">3.3 <span class="Heading">Creating [[6,42,,7,92127138,S1,
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B85A7217AF54FB1">3.3-1 NewPregroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78A45CA479E37F10">3.3-2 PregroupPresentationFromFp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D9C7247794CA029">3.3-3 PregroupPresentationToFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span java.lang.StringIndexOutOfBoundsException: Range [0, 55) out of bounds for length 44
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8194DE3E7908C921">3.4-1 IsPregroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84191A7B78073D28">3.4-2 </a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E53DCE97DEEFA7D">3.5 <span class="Heading">Hyperbolicity testing for pregroup presentations</span></a>
</span> [36,49,65282,]"^~S^1"]java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C607CC18455D320">3.5-1 RSymTestOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83D3AC54833FF1CE">3.5-2 RSymTest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DB6545987479C4B">3.5-3 IsHyperbolic</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X801CA5197DF125C6">3.6 <span class="Heading">Input and Output of Pregroup Presentations</ [42,43,,7,32,169,224,98,]"S^2S)12]java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81F8BD2784FA17A5">3.6-1 PregroupPresentationToKBMAG</a></span>
<span class="ContSS"><br 7,41244298,S^112]
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86E791EB7FCD6FC7">3.6-3 PregroupPresentationFromStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84B44E0580AE5C2A">3.6-4 PregroupPresentationToSimpleStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B118D1D84350906">3.6-5 PregroupPresentationToFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X808C30267956C4B0">3.6-6 PregroupPresentationFromFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84C6E86D7D3EC053">3.6-7 PregroupPresentationToSimpleFile</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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