<Chapter><Heading> Generators and relators of groups</Heading> <Section><Heading> &nbsp;</Heading> 
<ManSection> <Func Name="CayleyGraphOfGroupDisplay" Arg="G,X"/> <Func Name="CayleyGraphOfGroupDisplay" Arg="G,X,str"/> <Description> <P/> Inputs a finite group <M>G</M> together with a subset <M>X</M> of <M>G</M>. 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 <M>str</M>="mozilla". <P/> The argument <M>G</M> can also be a finite set of elements in a (possibly infinite) group containing <M>X</M>. The edges of the graph are coloured according to which element of <M>X</M> they are labelled by. The list <M>X</M> corresponds to the list of colours [blue, red, green, yellow, brown, black] in that order. <P/> This function requires Graphviz software. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="IdentityAmongRelatorsDisplay" Arg="R,n"/> <Func Name="IdentityAmongRelatorsDisplay" Arg="R,n,str"/> <Description> <P/> Inputs a free <M>ZG</M>-resolution <M>R</M> and an integer <M>n</M>. 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 <M>str</M>="mozilla". (The resolution <M>R</M> should be reduced and, preferably, in dimension 1 it should correspond to a Cayley graph for <M>G</M>. ) <P/>This function uses GraphViz software. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutPeriodic.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutTopology.html</Link><LinkText>3</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="IsAspherical" Arg="F,R"/> <Description> <P/> Inputs a free group <M>F</M> and a set <M>R</M> of words in <M>F</M>. It performs a test on the 2-dimensional CW-space <M>K</M> associated to this presentation for the group <M>G=F/</M>&tlt;<M>R</M>&tgt;<M>^F</M>. <P/> The function returns "true" if <M>K</M> has trivial second homotopy group. In this case it prints: Presentation is aspherical. <P/> Otherwise it returns "fail" and prints: Presentation is NOT piece-wise Euclidean non-positively curved. (In this case <M>K</M> 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/> The function uses Polymake software. <P/><B>Examples:</B> <URL><Link>../tutorial/chap3.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap6.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutAspherical.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>4</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="PresentationOfResolution" Arg="R"/> <Description> <P/> Inputs at least two terms of a reduced <M>ZG</M>-resolution <M>R</M> and returns a record <M>P</M> with components <List> <Item> <M>P.freeGroup</M> is a free group <M>F</M>, </Item> <Item> <M>P.relators</M> is a list <M>S</M> of words in <M>F</M>, </Item> <Item> <M>P.gens</M> is a list of positive integers such that the <M>i</M>-th generator of the presentation corresponds to the group element R!.elts[P[i]] . </Item> </List> where <M>G</M> is isomorphic to <M>F</M> modulo the normal closure of <M>S</M>. This presentation for <M>G</M> corresponds to the 2-skeleton of the classifying CW-space from which <M>R</M> was constructed. The resolution <M>R</M> requires no contracting homotopy. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutPolytopes.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutSpaceGroup.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutTopology.html</Link><LinkText>4</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="TorsionGeneratorsAbelianGroup" Arg="G"/> <Description> <P/> Inputs an abelian group <M>G</M> and returns a generating set <M>[x_1, \ldots ,x_n]</M> where no pair of generators have coprime orders. <P/><B>Examples:</B> 
</Description> </ManSection> </Section> </Chapter>