  
  [1X17 [33X[0;0YGenerators and relators of groups[133X[101X
  
  
  [1X17.1 [33X[0;0Y [133X[101X
  
  [1X17.1-1 CayleyGraphOfGroupDisplay[101X
  
  [33X[1;0Y[29X[2XCayleyGraphOfGroupDisplay[102X( [3XG[103X, [3XX[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCayleyGraphOfGroupDisplay[102X( [3XG[103X, [3XX[103X, [3Xstr[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  finite  group  [22XG[122X  together  with a subset [22XX[122X of [22XG[122X. 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 [22Xstr[122X="mozilla".[133X
  
  [33X[0;0YThe argument [22XG[122X can also be a finite set of elements in a (possibly infinite)
  group  containing  [22XX[122X. The edges of the graph are coloured according to which
  element  of  [22XX[122X  they  are labelled by. The list [22XX[122X corresponds to the list of
  colours [blue, red, green, yellow, brown, black] in that order.[133X
  
  [33X[0;0YThis function requires Graphviz software.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X17.1-2 IdentityAmongRelatorsDisplay[101X
  
  [33X[1;0Y[29X[2XIdentityAmongRelatorsDisplay[102X( [3XR[103X, [3Xn[103X ) [32X function[133X
  [33X[1;0Y[29X[2XIdentityAmongRelatorsDisplay[102X( [3XR[103X, [3Xn[103X, [3Xstr[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  free  [22XZG[122X-resolution  [22XR[122X and an integer [22Xn[122X. 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
  [22Xstr[122X="mozilla".  (The  resolution  [22XR[122X  should  be  reduced and, preferably, in
  dimension 1 it should correspond to a Cayley graph for [22XG[122X. )[133X
  
  [33X[0;0YThis function uses GraphViz software.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap6.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutPeriodic.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutTopology.html[107X) [133X
  
  [1X17.1-3 IsAspherical[101X
  
  [33X[1;0Y[29X[2XIsAspherical[102X( [3XF[103X, [3XR[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  free group [22XF[122X and a set [22XR[122X of words in [22XF[122X. It performs a test on the
  2-dimensional  CW-space  [22XK[122X  associated  to  this  presentation for the group
  [22XG=F/[122X<[22XR[122X>[22X^F[122X.[133X
  
  [33X[0;0YThe  function returns "true" if [22XK[122X has trivial second homotopy group. In this
  case it prints: Presentation is aspherical.[133X
  
  [33X[0;0YOtherwise  it  returns  "fail"  and  prints:  Presentation is NOT piece-wise
  Euclidean non-positively curved. (In this case [22XK[122X 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.)[133X
  
  [33X[0;0YThe function uses Polymake software.[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap3.html[107X) ,  2  ([7X../tutorial/chap6.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutAspherical.html[107X) ,                            4
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X17.1-4 PresentationOfResolution[101X
  
  [33X[1;0Y[29X[2XPresentationOfResolution[102X( [3XR[103X ) [32X function[133X
  
  [33X[0;0YInputs  at least two terms of a reduced [22XZG[122X-resolution [22XR[122X and returns a record
  [22XP[122X with components[133X
  
  [30X    [33X[0;6Y[22XP.freeGroup[122X is a free group [22XF[122X,[133X
  
  [30X    [33X[0;6Y[22XP.relators[122X is a list [22XS[122X of words in [22XF[122X,[133X
  
  [30X    [33X[0;6Y[22XP.gens[122X  is a list of positive integers such that the [22Xi[122X-th generator of
        the presentation corresponds to the group element R!.elts[P[i]] .[133X
  
  [33X[0;0Ywhere [22XG[122X is isomorphic to [22XF[122X modulo the normal closure of [22XS[122X. This presentation
  for [22XG[122X corresponds to the 2-skeleton of the classifying CW-space from which [22XR[122X
  was constructed. The resolution [22XR[122X requires no contracting homotopy.[133X
  
  [33X[0;0Y[12XExamples:[112X   1  ([7X../tutorial/chap6.html[107X) ,  2  ([7X../tutorial/chap13.html[107X) ,  3
  ([7X../tutorial/chap14.html[107X) , 4 ([7X../www/SideLinks/About/aboutPolytopes.html[107X) ,
  5              ([7X../www/SideLinks/About/aboutSpaceGroup.html[107X) ,             6
  ([7X../www/SideLinks/About/aboutTopology.html[107X) [133X
  
  [1X17.1-5 TorsionGeneratorsAbelianGroup[101X
  
  [33X[1;0Y[29X[2XTorsionGeneratorsAbelianGroup[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0YInputs an abelian group [22XG[122X and returns a generating set [22X[x_1, ... ,x_n][122X where
  no pair of generators have coprime orders.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
