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19. Coxeter diagrams and graphs of groups

19. 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.

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.


 


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