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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap25.html">[Previous Chapter]</a>    <a href="chap27.html">[Next Chapter]</a>   </div>

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<p><a id="X79D0502085B6734A" name="X79D0502085B6734A"></a></p>
<div class="ChapSects"><a href="chap26.html#X79D0502085B6734A">26 <span class="Heading"> Coxeter diagrams and graphs of groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7CFDEEC07F15CF82">26.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X812E91C980E14D4B">26.1-1 CoxeterDiagramComponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X84D4832E7E3760E0">26.1-2 CoxeterDiagramDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7DD8F9477A6C6774">26.1-3 CoxeterDiagramDisplay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F0324DE7DE0C3DD">26.1-4 CoxeterDiagramFpArtinGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7D3000FC786DAD98">26.1-5 CoxeterDiagramFpCoxeterGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F07DF3D810DDD29">26.1-6 CoxeterDiagramIsSpherical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X812D81E47B3A02AF">26.1-7 CoxeterDiagramMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X79D53A8A7EDE1AE2">26.1-8 CoxeterSubDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F2AAF2D8587C1C2">26.1-9 CoxeterDiagramVertices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8263BCA07F627536">26.1-10 EvenSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7873F6DB7B54F892">26.1-11 GraphOfGroupsDisplay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F1BE9C0863FFC20">26.1-12 GraphOfResolutions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8130246E854BC5D9">26.1-13 GraphOfGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X79C7CCCB7E209648">26.1-14 GraphOfResolutionsDisplay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X862B9A55867E651A">26.1-15 GraphOfGroupsTest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X816674BE7D86FD22">26.1-16 TreeOfGroupsToContractibleGcomplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X823C20DC7BF463DA">26.1-17 TreeOfResolutionsToContractibleGcomplex</a></span>
</div></div>
</div>

<h3>26 <span class="Heading"> Coxeter diagrams and graphs of groups</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>26.1 <span class="Heading">  </span></h4>

<p><a id="X812E91C980E14D4B" name="X812E91C980E14D4B"></a></p>

<h5>26.1-1 CoxeterDiagramComponents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramComponents</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and returns a list <span class="SimpleMath">[D_1, ..., D_d]</span> of the maximal connected subgraphs <span class="SimpleMath">D_i</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X84D4832E7E3760E0" name="X84D4832E7E3760E0"></a></p>

<h5>26.1-2 CoxeterDiagramDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramDegree</code>( <var class="Arg">D</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and vertex <span class="SimpleMath">v</span>. It returns the degree of <span class="SimpleMath">v</span> (i.e. the number of edges incident with <span class="SimpleMath">v</span>).</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7DD8F9477A6C6774" name="X7DD8F9477A6C6774"></a></p>

<h5>26.1-3 CoxeterDiagramDisplay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramDisplay</code>( <var class="Arg">D</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">‣ CoxeterDiagramDisplay</code>( <var class="Arg">D</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and displays it 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>This function requires Graphviz software.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap9.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutNoncrossing.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">7</a></span> </p>

<p><a id="X7F0324DE7DE0C3DD" name="X7F0324DE7DE0C3DD"></a></p>

<h5>26.1-4 CoxeterDiagramFpArtinGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramFpArtinGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and returns the corresponding finitely presented Artin group.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">1</a></span> </p>

<p><a id="X7D3000FC786DAD98" name="X7D3000FC786DAD98"></a></p>

<h5>26.1-5 CoxeterDiagramFpCoxeterGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramFpCoxeterGroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and returns the corresponding finitely presented Coxeter group.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">1</a></span> </p>

<p><a id="X7F07DF3D810DDD29" name="X7F07DF3D810DDD29"></a></p>

<h5>26.1-6 CoxeterDiagramIsSpherical</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramIsSpherical</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and returns "true" if the associated Coxeter groups is finite, and returns "false" otherwise.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">1</a></span> </p>

<p><a id="X812D81E47B3A02AF" name="X812D81E47B3A02AF"></a></p>

<h5>26.1-7 CoxeterDiagramMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramMatrix</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and returns a matrix representation of it. The matrix is given as a function <span class="SimpleMath">DiagramMatrix(D)(i,j)</span> where <span class="SimpleMath">i,j</span> can range over the vertices.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X79D53A8A7EDE1AE2" name="X79D53A8A7EDE1AE2"></a></p>

<h5>26.1-8 CoxeterSubDiagram</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterSubDiagram</code>( <var class="Arg">D</var>, <var class="Arg">V</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and a subset <span class="SimpleMath">V</span> of its vertices. It returns the full sub-diagram of <span class="SimpleMath">D</span> with vertex set <span class="SimpleMath">V</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7F2AAF2D8587C1C2" name="X7F2AAF2D8587C1C2"></a></p>

<h5>26.1-9 CoxeterDiagramVertices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoxeterDiagramVertices</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> and returns its set of vertices.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X8263BCA07F627536" name="X8263BCA07F627536"></a></p>

<h5>26.1-10 EvenSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EvenSubgroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> and returns a subgroup <span class="SimpleMath">G^+</span>. The subgroup is that generated by all products <span class="SimpleMath">xy</span> where <span class="SimpleMath">x</span> and <span class="SimpleMath">y</span> range over the generating set for <span class="SimpleMath">G</span> stored by GAP. The subgroup is probably only meaningful when <span class="SimpleMath">G</span> is an Artin or Coxeter group.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">2</a></span> </p>

<p><a id="X7873F6DB7B54F892" name="X7873F6DB7B54F892"></a></p>

<h5>26.1-11 GraphOfGroupsDisplay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GraphOfGroupsDisplay</code>( <var class="Arg">D</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">‣ GraphOfGroupsDisplay</code>( <var class="Arg">D</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of groups <span class="SimpleMath">D</span> and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using the second argument <span class="SimpleMath">str</span>="mozilla".</p>

<p>This function requires Graphviz software.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGraphsOfGroups.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">4</a></span> </p>

<p><a id="X7F1BE9C0863FFC20" name="X7F1BE9C0863FFC20"></a></p>

<h5>26.1-12 GraphOfResolutions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GraphOfResolutions</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of groups <span class="SimpleMath">D</span> and a positive integer <span class="SimpleMath">n</span>. It returns a graph of resolutions, each resolution being of length <span class="SimpleMath">n</span>. It uses the function ResolutionGenericGroup() to produce the resolutions.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X8130246E854BC5D9" name="X8130246E854BC5D9"></a></p>

<h5>26.1-13 GraphOfGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GraphOfGroups</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of resolutions <span class="SimpleMath">D</span> and returns the corresponding graph of groups.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGraphsOfGroups.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">4</a></span> </p>

<p><a id="X79C7CCCB7E209648" name="X79C7CCCB7E209648"></a></p>

<h5>26.1-14 GraphOfResolutionsDisplay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GraphOfResolutionsDisplay</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of resolutions <span class="SimpleMath">D</span> and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram.</p>

<p>This function requires Graphviz software.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X862B9A55867E651A" name="X862B9A55867E651A"></a></p>

<h5>26.1-15 GraphOfGroupsTest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GraphOfGroupsTest</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object <span class="SimpleMath">D</span> and itries to test whether it is a Graph of Groups. However, it DOES NOT test the injectivity of any homomorphisms. It returns true if <span class="SimpleMath">D</span> passes the test, and false otherwise.</p>

<p>Note that there is no function <span class="SimpleMath">IsHapGraphOfGroups()</span> because no special data type has been created for these graphs.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X816674BE7D86FD22" name="X816674BE7D86FD22"></a></p>

<h5>26.1-16 TreeOfGroupsToContractibleGcomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TreeOfGroupsToContractibleGcomplex</code>( <var class="Arg">D</var>, <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of groups <span class="SimpleMath">D</span> which is a tree, and also inputs the fundamental group <span class="SimpleMath">G</span> of the tree in a form which contains each of the groups in the graph as subgroups. It returns a corresponding contractible G-complex.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X823C20DC7BF463DA" name="X823C20DC7BF463DA"></a></p>

<h5>26.1-17 TreeOfResolutionsToContractibleGcomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TreeOfResolutionsToContractibleGcomplex</code>( <var class="Arg">D</var>, <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of resolutions <span class="SimpleMath">D</span> which is a tree, and also inputs the fundamental group <span class="SimpleMath">G</span> of the tree in a form which contains each of the groups in the graph as subgroups. It returns a corresponding contractible G-complex. The resolutions are stored as a component of the contractible <span class="SimpleMath">G</span>-complex.</p>

<p><strong class="button">Examples:</strong></p>


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