<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 <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>
<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, ... ,x_n]</span> where no pair of generators have coprime orders.</p>
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