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<div class="ChapSects"><a href="chap17_mj.html#X7A2144518112F830">17 <span class="Heading"> Generators and relators of groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17_mj.html#X7CFDEEC07F15CF82">17.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X7F49A86A82EB2420">17.1-1 CayleyGraphOfGroupDisplay</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X82C8B87287602BFA">17.1-2 IdentityAmongRelatorsDisplay</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X78F2C5ED80D1C8DD">17.1-3 IsAspherical</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X878938C3835871D7">17.1-4 PresentationOfResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap17_mj.html#X7F71698178AF48DD">17.1-5 TorsionGeneratorsAbelianGroup</a></span>
</div></div>
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<h3>17 <span class="Heading"> Generators and relators of groups</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>17.1 <span class="Heading"> </span></h4>
<p><a id="X7F49A86A82EB2420" name="X7F49A86A82EB2420"></a></p>
<h5>17.1-1 CayleyGraphOfGroupDisplay</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CayleyGraphOfGroupDisplay</code>( <var class="Arg">G</var>, <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CayleyGraphOfGroupDisplay</code>( <var class="Arg">G</var>, <var class="Arg">X</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span> together with a subset <span class="SimpleMath">\(X\)</span> of <span class="SimpleMath">\(G\)</span>. It displays the corresponding Cayley graph as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using a second argument <span class="SimpleMath">\(str\)</span>="mozilla".</p>
<p>The argument <span class="SimpleMath">\(G\)</span> can also be a finite set of elements in a (possibly infinite) group containing <span class="SimpleMath">\(X\)</span>. The edges of the graph are coloured according to which element of <span class="SimpleMath">\(X\)</span> they are labelled by. The list <span class="SimpleMath">\(X\)</span> corresponds to the list of colours [blue, red, green, yellow, brown, black] in that order.</p>
<p>This function requires Graphviz software.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X82C8B87287602BFA" name="X82C8B87287602BFA"></a></p>
<h5>17.1-2 IdentityAmongRelatorsDisplay</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IdentityAmongRelatorsDisplay</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IdentityAmongRelatorsDisplay</code>( <var class="Arg">R</var>, <var class="Arg">n</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> and an integer <span class="SimpleMath">\(n\)</span>. It displays the boundary R!.boundary(3,n) as a tessellation of a sphere. It displays the tessellation as a .gif file and uses the Mozilla web browser as a default display mechanism. An alternative browser can be set using the second argument <span class="SimpleMath">\(str\)</span>="mozilla". (The resolution <span class="SimpleMath">\(R\)</span> should be reduced and, preferably, in dimension 1 it should correspond to a Cayley graph for <span class="SimpleMath">\(G\)</span>. )</p>
<p>This function uses GraphViz software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPeriodic.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTopology.html">3</a></span> </p>
<p><a id="X78F2C5ED80D1C8DD" name="X78F2C5ED80D1C8DD"></a></p>
<h5>17.1-3 IsAspherical</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsAspherical</code>( <var class="Arg">F</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free group <span class="SimpleMath">\(F\)</span> and a set <span class="SimpleMath">\(R\)</span> of words in <span class="SimpleMath">\(F\)</span>. It performs a test on the 2-dimensional CW-space <span class="SimpleMath">\(K\)</span> associated to this presentation for the group <span class="SimpleMath">\(G=F/\)</span><<span class="SimpleMath">\(R\)</span>><span class="SimpleMath">\(^F\)</span>.</p>
<p>The function returns "true" if <span class="SimpleMath">\(K\)</span> has trivial second homotopy group. In this case it prints: Presentation is aspherical.</p>
<p>Otherwise it returns "fail" and prints: Presentation is NOT piece-wise Euclidean non-positively curved. (In this case <span class="SimpleMath">\(K\)</span> may or may not have trivial second homotopy group. But it is NOT possible to impose a metric on K which restricts to a Euclidean metric on each 2-cell.)</p>
<p>The function uses Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap3.html">1</a></span> , <span class="URL"><a href="../tutorial/chap6.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAspherical.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">4</a></span> </p>
<p><a id="X878938C3835871D7" name="X878938C3835871D7"></a></p>
<h5>17.1-4 PresentationOfResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PresentationOfResolution</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs at least two terms of a reduced <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> and returns a record <span class="SimpleMath">\(P\)</span> with components</p>
<ul>
<li><p><span class="SimpleMath">\(P.freeGroup\)</span> is a free group <span class="SimpleMath">\(F\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(P.relators\)</span> is a list <span class="SimpleMath">\(S\)</span> of words in <span class="SimpleMath">\(F\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(P.gens\)</span> is a list of positive integers such that the <span class="SimpleMath">\(i\)</span>-th generator of the presentation corresponds to the group element R!.elts[P[i]] .</p>
</li>
</ul>
<p>where <span class="SimpleMath">\(G\)</span> is isomorphic to <span class="SimpleMath">\(F\)</span> modulo the normal closure of <span class="SimpleMath">\(S\)</span>. This presentation for <span class="SimpleMath">\(G\)</span> corresponds to the 2-skeleton of the classifying CW-space from which <span class="SimpleMath">\(R\)</span> was constructed. The resolution <span class="SimpleMath">\(R\)</span> requires no contracting homotopy.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../tutorial/chap14.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTopology.html">6</a></span> </p>
<p><a id="X7F71698178AF48DD" name="X7F71698178AF48DD"></a></p>
<h5>17.1-5 TorsionGeneratorsAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TorsionGeneratorsAbelianGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an abelian group <span class="SimpleMath">\(G\)</span> and returns a generating set <span class="SimpleMath">\([x_1, \ldots ,x_n]\)</span> where no pair of generators have coprime orders.</p>
<p><strong class="button">Examples:</strong></p>
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