  
  [1X26 [33X[0;0YCoxeter diagrams and graphs of groups[133X[101X
  
  
  [1X26.1 [33X[0;0Y [133X[101X
  
  [1X26.1-1 CoxeterDiagramComponents[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramComponents[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs a Coxeter diagram [22XD[122X and returns a list [22X[D_1, ..., D_d][122X of the maximal
  connected subgraphs [22XD_i[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-2 CoxeterDiagramDegree[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramDegree[102X( [3XD[103X, [3Xv[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  Coxeter  diagram [22XD[122X and vertex [22Xv[122X. It returns the degree of [22Xv[122X (i.e.
  the number of edges incident with [22Xv[122X).[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-3 CoxeterDiagramDisplay[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramDisplay[102X( [3XD[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCoxeterDiagramDisplay[102X( [3XD[103X, [3Xstr[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  Coxeter  diagram  [22XD[122X  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 [22Xstr[122X="mozilla".[133X
  
  [33X[0;0YThis function requires Graphviz software.[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap7.html[107X) ,  2  ([7X../tutorial/chap9.html[107X) ,  3
  ([7X../tutorial/chap11.html[107X) ,                                                4
  ([7X../www/SideLinks/About/aboutArtinGroups.html[107X) ,                           5
  ([7X../www/SideLinks/About/aboutNoncrossing.html[107X) ,                           6
  ([7X../www/SideLinks/About/aboutPolytopes.html[107X) ,                             7
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X26.1-4 CoxeterDiagramFpArtinGroup[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramFpArtinGroup[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a Coxeter diagram [22XD[122X and returns the corresponding finitely presented
  Artin group.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutArtinGroups.html[107X) [133X
  
  [1X26.1-5 CoxeterDiagramFpCoxeterGroup[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramFpCoxeterGroup[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a Coxeter diagram [22XD[122X and returns the corresponding finitely presented
  Coxeter group.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutArtinGroups.html[107X) [133X
  
  [1X26.1-6 CoxeterDiagramIsSpherical[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramIsSpherical[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  Coxeter  diagram  [22XD[122X  and returns "true" if the associated Coxeter
  groups is finite, and returns "false" otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutArtinGroups.html[107X) [133X
  
  [1X26.1-7 CoxeterDiagramMatrix[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramMatrix[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  Coxeter  diagram [22XD[122X and returns a matrix representation of it. The
  matrix is given as a function [22XDiagramMatrix(D)(i,j)[122X where [22Xi,j[122X can range over
  the vertices.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-8 CoxeterSubDiagram[101X
  
  [33X[1;0Y[29X[2XCoxeterSubDiagram[102X( [3XD[103X, [3XV[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  Coxeter  diagram [22XD[122X and a subset [22XV[122X of its vertices. It returns the
  full sub-diagram of [22XD[122X with vertex set [22XV[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-9 CoxeterDiagramVertices[101X
  
  [33X[1;0Y[29X[2XCoxeterDiagramVertices[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs a Coxeter diagram [22XD[122X and returns its set of vertices.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-10 EvenSubgroup[101X
  
  [33X[1;0Y[29X[2XEvenSubgroup[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0YInputs  a group [22XG[122X and returns a subgroup [22XG^+[122X. The subgroup is that generated
  by  all products [22Xxy[122X where [22Xx[122X and [22Xy[122X range over the generating set for [22XG[122X stored
  by  GAP.  The  subgroup  is  probably  only meaningful when [22XG[122X is an Artin or
  Coxeter group.[133X
  
  [33X[0;0Y[12XExamples:[112X      1      ([7X../www/SideLinks/About/aboutArtinGroups.html[107X) ,     2
  ([7X../www/SideLinks/About/aboutTwistedCoefficients.html[107X) [133X
  
  [1X26.1-11 GraphOfGroupsDisplay[101X
  
  [33X[1;0Y[29X[2XGraphOfGroupsDisplay[102X( [3XD[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGraphOfGroupsDisplay[102X( [3XD[103X, [3Xstr[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  graph  of  groups  [22XD[122X  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 [22Xstr[122X="mozilla".[133X
  
  [33X[0;0YThis function requires Graphviz software.[133X
  
  [33X[0;0Y[12XExamples:[112X   1  ([7X../tutorial/chap7.html[107X) ,  2  ([7X../tutorial/chap11.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutGraphsOfGroups.html[107X) ,                        4
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X26.1-12 GraphOfResolutions[101X
  
  [33X[1;0Y[29X[2XGraphOfResolutions[102X( [3XD[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  graph of groups [22XD[122X and a positive integer [22Xn[122X. It returns a graph of
  resolutions,  each  resolution  being  of  length  [22Xn[122X.  It  uses the function
  ResolutionGenericGroup() to produce the resolutions.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-13 GraphOfGroups[101X
  
  [33X[1;0Y[29X[2XGraphOfGroups[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  graph  of  resolutions  [22XD[122X  and returns the corresponding graph of
  groups.[133X
  
  [33X[0;0Y[12XExamples:[112X   1  ([7X../tutorial/chap7.html[107X) ,  2  ([7X../tutorial/chap11.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutGraphsOfGroups.html[107X) ,                        4
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X26.1-14 GraphOfResolutionsDisplay[101X
  
  [33X[1;0Y[29X[2XGraphOfResolutionsDisplay[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a graph of resolutions [22XD[122X and displays it as a .gif file. It uses the
  Mozilla web browser as a default to view the diagram.[133X
  
  [33X[0;0YThis function requires Graphviz software.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-15 GraphOfGroupsTest[101X
  
  [33X[1;0Y[29X[2XGraphOfGroupsTest[102X( [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  an  object  [22XD[122X  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 [22XD[122X passes the test, and false otherwise.[133X
  
  [33X[0;0YNote  that there is no function [22XIsHapGraphOfGroups()[122X because no special data
  type has been created for these graphs.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-16 TreeOfGroupsToContractibleGcomplex[101X
  
  [33X[1;0Y[29X[2XTreeOfGroupsToContractibleGcomplex[102X( [3XD[103X, [3XG[103X ) [32X function[133X
  
  [33X[0;0YInputs  a graph of groups [22XD[122X which is a tree, and also inputs the fundamental
  group [22XG[122X of the tree in a form which contains each of the groups in the graph
  as subgroups. It returns a corresponding contractible G-complex.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X26.1-17 TreeOfResolutionsToContractibleGcomplex[101X
  
  [33X[1;0Y[29X[2XTreeOfResolutionsToContractibleGcomplex[102X( [3XD[103X, [3XG[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  graph  of  resolutions  [22XD[122X  which  is  a tree, and also inputs the
  fundamental  group [22XG[122X 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
  [22XG[122X-complex.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
