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<p><a id="X7A05461687E1FA84" name="X7A05461687E1FA84"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X7A05461687E1FA84">3 <span class="Heading">Pregroup Presentations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.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_mj.html#X83BEFE747F7726F5">3.1-1 <span class="Heading">Locations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87FA460D81566861">3.1-2 <span class="Heading">Places</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.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_mj.html#X81378EEB81DF801A">3.2-1 IsPregroupLocation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85EC64DC8116AF77">3.2-2 InLetter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E1013A1794D17DD">3.2-3 OutLetter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81E1F7F984EC37C2">3.2-4 Places</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C88C1227C073A36">3.2-5 NextLocation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83B0E7F778108E23">3.2-6 PrevLocation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X796A41AD85E9AD40">3.2-7 __ID</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X85C45E1578DB9E13">3.3 <span class="Heading">Creating Pregroup Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B85A7217AF54FB1">3.3-1 NewPregroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78A45CA479E37F10">3.3-2 PregroupPresentationFromFp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D9C7247794CA029">3.3-3 PregroupPresentationToFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7FC6BCF985E6819C">3.4 <span class="Heading">Filters, Attributes, and Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8194DE3E7908C921">3.4-1 IsPregroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84191A7B78073D28">3.4-2 </a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7E53DCE97DEEFA7D">3.5 <span class="Heading">Hyperbolicity testing for pregroup presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C607CC18455D320">3.5-1 RSymTestOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83D3AC54833FF1CE">3.5-2 RSymTest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DB6545987479C4B">3.5-3 IsHyperbolic</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X801CA5197DF125C6">3.6 <span class="Heading">Input and Output of Pregroup Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81F8BD2784FA17A5">3.6-1 PregroupPresentationToKBMAG</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X844A431F8505219D">3.6-2 PregroupPresentationToStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86E791EB7FCD6FC7">3.6-3 PregroupPresentationFromStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84B44E0580AE5C2A">3.6-4 PregroupPresentationToSimpleStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B118D1D84350906">3.6-5 PregroupPresentationToFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X808C30267956C4B0">3.6-6 PregroupPresentationFromFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84C6E86D7D3EC053">3.6-7 PregroupPresentationToSimpleFile</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Pregroup Presentations</span></h3>

<p><a id="X7AFDBDC078777B90" name="X7AFDBDC078777B90"></a></p>

<h4>3.1 <span class="Heading">Concepts</span></h4>

<p>Given a pregroup <span class="SimpleMath">\(P\)</span> there is a universal group <span class="SimpleMath">\(\mathcal{U}(P)\)</span> that contains <span class="SimpleMath">\(P\)</span>. The concept of a pregroup presentation is a generalisation of presentations over the free group, that is a pregroup presentation is a way of defining a group as a quotient of a universal group over a pregroup by giving relator words over the pregroup.</p>

<p>For the purposes of the RSym tester we introduce some more concepts.</p>

<p><a id="X83BEFE747F7726F5" name="X83BEFE747F7726F5"></a></p>

<h5>3.1-1 <span class="Heading">Locations</span></h5>

<p>A <em>location</em> on a pregroup relator <span class="SimpleMath">\( w = a_1a_2\ldots a_n\)</span> is an index <span class="SimpleMath">\(i\)</span> between <span class="SimpleMath">\(1\)</span> and <span class="SimpleMath">\(n\)</span> and denotes the location between <span class="SimpleMath">\(a_i\)</span> (the <code class="func">InLetter</code> (<a href="chap3_mj.html#X85EC64DC8116AF77"><span class="RefLink">3.2-2</span></a>)) and <span class="SimpleMath">\(a_{i+1}\)</span> (the <code class="func">OutLetter</code> (<a href="chap3_mj.html#X7E1013A1794D17DD"><span class="RefLink">3.2-3</span></a>)), where the relator is considered cyclically, that is, when <span class="SimpleMath">\(i=n\)</span> then the outletter is <span class="SimpleMath">\(a_1\)</span>.</p>

<p><a id="X87FA460D81566861" name="X87FA460D81566861"></a></p>

<h5>3.1-2 <span class="Heading">Places</span></h5>

<p>A <em>place</em> <span class="SimpleMath">\(R(L, x, C)\)</span> on a pregroup relator <span class="SimpleMath">\(R\)</span> is a location (<a href="chap3_mj.html#X83BEFE747F7726F5"><span class="RefLink">3.1-1</span></a>) together with a letter from the pregroup and a colour, which is either <em>red</em> or <em>green</em>.</p>

<p><a id="X7C701DBF7BAE649A" name="X7C701DBF7BAE649A"></a></p>

<h4>3.2 <span class="Heading">Attributes</span></h4>

<p><a id="X81378EEB81DF801A" name="X81378EEB81DF801A"></a></p>

<h5>3.2-1 IsPregroupLocation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPregroupLocation</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p><a id="X85EC64DC8116AF77" name="X85EC64DC8116AF77"></a></p>

<h5>3.2-2 InLetter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InLetter</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X7E1013A1794D17DD" name="X7E1013A1794D17DD"></a></p>

<h5>3.2-3 OutLetter</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OutLetter</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X81E1F7F984EC37C2" name="X81E1F7F984EC37C2"></a></p>

<h5>3.2-4 Places</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Places</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X7C88C1227C073A36" name="X7C88C1227C073A36"></a></p>

<h5>3.2-5 NextLocation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NextLocation</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X83B0E7F778108E23" name="X83B0E7F778108E23"></a></p>

<h5>3.2-6 PrevLocation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrevLocation</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X796A41AD85E9AD40" name="X796A41AD85E9AD40"></a></p>

<h5>3.2-7 __ID</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ __ID</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X85C45E1578DB9E13" name="X85C45E1578DB9E13"></a></p>

<h4>3.3 <span class="Heading">Creating Pregroup Presentations</span></h4>

<p><a id="X7B85A7217AF54FB1" name="X7B85A7217AF54FB1"></a></p>

<h5>3.3-1 NewPregroupPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NewPregroupPresentation</code>( <var class="Arg">pregroup</var>, <var class="Arg">relators</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a pregroup presentation</p>

<p>Creates a pregroup presentation over the <var class="Arg">pregroup</var> with relators <var class="Arg">relators</var>.</p>

<p><a id="X78A45CA479E37F10" name="X78A45CA479E37F10"></a></p>

<h5>3.3-2 PregroupPresentationFromFp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationFromFp</code>( <var class="Arg">F</var>, <var class="Arg">rred</var>, <var class="Arg">rgreen</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a pregroup presentation</p>

<p>Creates a pregroup presentation over the pregroup defined by <var class="Arg">F</var> and <var class="Arg">rred</var> with relators <var class="Arg">rgreen</var>.</p>

<p><a id="X7D9C7247794CA029" name="X7D9C7247794CA029"></a></p>

<h5>3.3-3 PregroupPresentationToFpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationToFpGroup</code>( <var class="Arg">presentation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a finitely presented group</p>

<p>Converts the pregroup presentation <var class="Arg">presentation</var> into a finitely presented group.</p>

<p><a id="X7FC6BCF985E6819C" name="X7FC6BCF985E6819C"></a></p>

<h4>3.4 <span class="Heading">Filters, Attributes, and Properties</span></h4>

<p><a id="X8194DE3E7908C921" name="X8194DE3E7908C921"></a></p>

<h5>3.4-1 IsPregroupPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPregroupPresentation</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p><a id="X84191A7B78073D28" name="X84191A7B78073D28"></a></p>

<h5>3.4-2 </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ </code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p><a id="X7E53DCE97DEEFA7D" name="X7E53DCE97DEEFA7D"></a></p>

<h4>3.5 <span class="Heading">Hyperbolicity testing for pregroup presentations</span></h4>

<p><a id="X7C607CC18455D320" name="X7C607CC18455D320"></a></p>

<h5>3.5-1 RSymTestOp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RSymTestOp</code>( <var class="Arg">presentation</var>, <var class="Arg">epsilon</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Test the group presented by <var class="Arg">presentation</var> for hyperbolicity using the RSym tester with parameter <var class="Arg">epsilon</var>.</p>

<p><a id="X83D3AC54833FF1CE" name="X83D3AC54833FF1CE"></a></p>

<h5>3.5-2 RSymTest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RSymTest</code>( <var class="Arg">args...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is a wrapper for <code class="func">RSymTestOp</code> (<a href="chap3_mj.html#X7C607CC18455D320"><span class="RefLink">3.5-1</span></a>). If the first argument given is a free group, the second and third lists of words over the free group, and the fourth a rational, then this function creates a pregroup presentation from the input data and invokes <code class="func">RSymTestOp</code> (<a href="chap3_mj.html#X7C607CC18455D320"><span class="RefLink">3.5-1</span></a>) on it. If the first argument is a pregroup presentation and the second argument is rational number, then it invokes <code class="func">RSymTestOp</code> (<a href="chap3_mj.html#X7C607CC18455D320"><span class="RefLink">3.5-1</span></a>) on that input.</p>

<p><a id="X7DB6545987479C4B" name="X7DB6545987479C4B"></a></p>

<h5>3.5-3 IsHyperbolic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHyperbolic</code>( <var class="Arg">presentation</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">‣ IsHyperbolic</code>( <var class="Arg">presentation</var>, <var class="Arg">epsilon</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">‣ IsHyperbolic</code>( <var class="Arg">F</var>, <var class="Arg">rred</var>, <var class="Arg">rgreen</var>, <var class="Arg">epsilon</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Tests a given presentation for hyperbolicity using the RSym test procedure.</p>

<p><a id="X801CA5197DF125C6" name="X801CA5197DF125C6"></a></p>

<h4>3.6 <span class="Heading">Input and Output of Pregroup Presentations</span></h4>

<p><a id="X81F8BD2784FA17A5" name="X81F8BD2784FA17A5"></a></p>

<h5>3.6-1 PregroupPresentationToKBMAG</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationToKBMAG</code>( <var class="Arg">presentation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A KBMAG rewriting system</p>

<p>Turns the pregroup presentation <var class="Arg">presentation</var> into valid input for Knuth-Bendix rewriting using KBMAG. Only available if the kbmag package is available.</p>

<p><a id="X844A431F8505219D" name="X844A431F8505219D"></a></p>

<h5>3.6-2 PregroupPresentationToStream</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationToStream</code>( <var class="Arg">stream</var>, <var class="Arg">presentation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Writes the pregroup presentation <var class="Arg">presentation</var> to <var class="Arg">stream</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T := TriangleGroup(2,3,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">str := "";; stream := OutputTextString(str, true);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PregroupPresentationToStream(stream, T);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print(str);</span>
rec(
  rels := [ [ 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3 ] ],
  table := [ [ 1, 2, 3, 4 ], [ 2, 1, 0, 0 ], [ 3, 0, 4, 1 ], [ 4, 0, 1, 3 ] ] );
</pre></div>

<p><a id="X86E791EB7FCD6FC7" name="X86E791EB7FCD6FC7"></a></p>

<h5>3.6-3 PregroupPresentationFromStream</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationFromStream</code>( <var class="Arg">stream</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A pregroup presentation</p>

<p>Reads a pregroup presentation from an input stream in the same format that <code class="func">PregroupPresentationToStream</code> (<a href="chap3_mj.html#X844A431F8505219D"><span class="RefLink">3.6-2</span></a>) uses.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">stream := InputTextString(str);</span>
InputTextString(0,146)
<span class="GAPprompt">gap></span> <span class="GAPinput">PregroupPresentationFromStream(stream);</span>
<pregroup presentation with 3 generators and 1 relators>
</pre></div>

<p><a id="X84B44E0580AE5C2A" name="X84B44E0580AE5C2A"></a></p>

<h5>3.6-4 PregroupPresentationToSimpleStream</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationToSimpleStream</code>( <var class="Arg">stream</var>, <var class="Arg">presentation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Writes the pregroup presentation <var class="Arg">presentation</var> to <var class="Arg">stream</var>. Uses a simpler format than <code class="func">PregroupPresentationToStream</code> (<a href="chap3_mj.html#X844A431F8505219D"><span class="RefLink">3.6-2</span></a>)</p>

<p><a id="X7B118D1D84350906" name="X7B118D1D84350906"></a></p>

<h5>3.6-5 PregroupPresentationToFile</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationToFile</code>( <var class="Arg">filename</var>, <var class="Arg">presentation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Writes the pregroup presentation <var class="Arg">presentation</var> to file with name <var class="Arg">filename</var>.</p>

<p><a id="X808C30267956C4B0" name="X808C30267956C4B0"></a></p>

<h5>3.6-6 PregroupPresentationFromFile</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationFromFile</code>( <var class="Arg">filename</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Reads a pregroup presentation from file with <var class="Arg">filename</var>.</p>

<p><a id="X84C6E86D7D3EC053" name="X84C6E86D7D3EC053"></a></p>

<h5>3.6-7 PregroupPresentationToSimpleFile</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PregroupPresentationToSimpleFile</code>( <var class="Arg">stream</var>, <var class="Arg">presentation</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Writes the pregroup presentation <var class="Arg">presentation</var> to file with name <var class="Arg">filename</var> in a simple format.</p>


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