<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 <spanclass="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 <spanclass="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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
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