GAP (HAP) - Chapter 22: Coxeter diagrams and graphs of groups

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22 Coxeter diagrams and graphs of groups

### 22 Coxeter diagrams and graphs of groups

`CoxeterDiagramComponents(D)` Inputs a Coxeter diagram D and returns a list [D_1, ..., D_d] of the maximal connected subgraphs D_i.
`CoxeterDiagramDegree(D,v)` Inputs a Coxeter diagram D and vertex v. It returns the degree of v (i.e. the number of edges incident with v).
`CoxeterDiagramDisplay(D)` `CoxeterDiagramDisplay(D,"web browser")` Inputs a Coxeter diagram D 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. This function requires Graphviz software.
`CoxeterDiagramFpArtinGroup(D)` Inputs a Coxeter diagram D and returns the corresponding finitely presented Artin group.
`CoxeterDiagramFpCoxeterGroup(D)` Inputs a Coxeter diagram D and returns the corresponding finitely presented Coxeter group.
`CoxeterDiagramIsSpherical(D)` Inputs a Coxeter diagram D and returns "true" if the associated Coxeter groups is finite, and returns "false" otherwise.
`CoxeterDiagramMatrix(D)` Inputs a Coxeter diagram D and returns a matrix representation of it. The matrix is given as a function DiagramMatrix(D)(i,j) where i,j can range over the vertices.
`CoxeterSubDiagram(D,V)` Inputs a Coxeter diagram D and a subset V of its vertices. It returns the full sub-diagram of D with vertex set V.
`CoxeterDiagramVertices(D)` Inputs a Coxeter diagram D and returns its set of vertices.
`EvenSubgroup(G)` Inputs a group G and returns a subgroup G^+. The subgroup is that generated by all products xy where x and y range over the generating set for G stored by GAP. The subgroup is probably only meaningful when G is an Artin or Coxeter group.
`GraphOfGroupsDisplay(D)` `GraphOfGroupsDisplay(D,"web browser")` Inputs a graph of groups D 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. This function requires Graphviz software.
`GraphOfResolutions(D,n)` Inputs a graph of groups D and a positive integer n. It returns a graph of resolutions, each resolution being of length n. It uses the function ResolutionGenericGroup() to produce the resolutions.
`GraphOfGroups(D)` Inputs a graph of resolutions D and returns the corresponding graph of groups.
`GraphOfResolutionsDisplay(D)` Inputs a graph of resolutions D and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram. This function requires Graphviz software.
`GraphOfGroupsTest(D)` Inputs an object D 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 D passes the test, and false otherwise. Note that there is no function IsHapGraphOfGroups() because no special data type has been created for these graphs.
`TreeOfGroupsToContractibleGcomplex(D,G)` Inputs a graph of groups D which is a tree, and also inputs the fundamental group G of the tree in a form which contains each of the groups in the graph as subgroups. It returns a corresponding contractible G-complex.
`TreeOfResolutionsToContractibleGcomplex(D,G)` Inputs a graph of resolutions D which is a tree, and also inputs the fundamental group G 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 G-complex.

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