<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (HAP) - Chapter 11: Generators and relators of groups</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
</head>
<body><a href="../www/index.html"><small>HAP home</small></a>
<div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap10.html">Previous Chapter</a> <a href="chap12.html">Next Chapter</a> </div>
<p><a id="X7A2144518112F830" name="X7A2144518112F830"></a></p>
<div class="ChapSects"><a href="chap11.html#X7A2144518112F830">11. <span class="Heading"> Generators and relators of groups</span></a>
</div>
<h3>11. <span class="Heading"> Generators and relators of groups</span></h3>
<div class="pcenter"><table cellspacing="10" class="GAPDocTable">
<tr>
<td class="tdleft"><code class="code"> CayleyGraphDisplay(G,X) </code> <br /> <code class="code"> CayleyGraphDisplay(G,X,"mozilla") </code></p>
<p>Inputs a finite group G together with a subset X of G. 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.</p>
<p>The argument G can also be a finite set of elements in a (possibly infinite) group containing X. The edges of the graph are coloured according to which element of X they are labelled by. The list X corresponds to the list of colours [blue, red, green, yellow, brown, black] in that order.</p>
<p>This function requires Graphviz software.</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> IdentityAmongRelatorsDisplay(R,n) </code> <code class="code"> IdentityAmongRelatorsDisplay(R,n,"mozilla") </code></p>
<p>Inputs a free ZG-resolution R and an integer n. 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 a second argument. (The resolution R should be reduced and, preferably, in dimension 1 it should correspond to a Cayley graph for G. )</p>
<p>This function uses GraphViz software.</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> IsAspherical(F,R) </code></p>
<p>Inputs a free group F and a set R of words in F. It performs a test on the 2-dimensional CW-space K associated to this presentation for the group G=F/<R>^F.</p>
<p>The function returns "true" if K 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 K 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.</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> PresentationOfResolution(R) </code></p>
<p>Inputs at least two terms of a reduced ZG-resolution R and returns a record P with components</p>
<ul>
<li><p>P.freeGroup is a free group F,</p>
</li>
<li><p>P.relators is a list S of words in F,</p>
</li>
</ul>
<p>where G is isomorphic to F modulo the normal closure of S. This presentation for G corresponds to the 2-skeleton of the classifying CW-space from which R was constructed. The resolution R requires no contracting homotopy.</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> TorsionGeneratorsAbelianGroup(G) </code></p>
<p>Inputs an abelian group G and returns a generating set [x_1, ... ,x_n] where no pair of generators have coprime orders.</td>
</tr>
</table><br /><p> </p><br />
</div>
<div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap10.html">Previous Chapter</a> <a href="chap12.html">Next Chapter</a> </div>
<div class="chlinkbot"><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="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
distrib
>
Mandriva
>
2010.0
>
i586
>
media
>
contrib-release
>
by-pkgid
>
91213ddcfbe7f54821d42c2d9e091326
>
files
>
163
