GAP (HAP commands) - Chapter 26: Coxeter diagrams and graphs of groups

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26 Coxeter diagrams and graphs of groups
26.1
26.1-1 CoxeterDiagramComponents
26.1-2 CoxeterDiagramDegree
26.1-3 CoxeterDiagramDisplay
26.1-4 CoxeterDiagramFpArtinGroup
26.1-5 CoxeterDiagramFpCoxeterGroup
26.1-6 CoxeterDiagramIsSpherical
26.1-7 CoxeterDiagramMatrix
26.1-8 CoxeterSubDiagram
26.1-9 CoxeterDiagramVertices
26.1-10 EvenSubgroup
26.1-11 GraphOfGroupsDisplay
26.1-12 GraphOfResolutions
26.1-13 GraphOfGroups
26.1-14 GraphOfResolutionsDisplay
26.1-15 GraphOfGroupsTest
26.1-16 TreeOfGroupsToContractibleGcomplex
26.1-17 TreeOfResolutionsToContractibleGcomplex

### 26 Coxeter diagrams and graphs of groups

#### 26.1

##### 26.1-1 CoxeterDiagramComponents

`‣ CoxeterDiagramComponents`( D ) | ( function )

Inputs a Coxeter diagram D and returns a list [D_1, ..., D_d] of the maximal connected subgraphs D_i.

Examples:

##### 26.1-2 CoxeterDiagramDegree

`‣ CoxeterDiagramDegree`( D, v ) | ( function )

Inputs a Coxeter diagram D and vertex v. It returns the degree of v (i.e. the number of edges incident with v).

Examples:

##### 26.1-3 CoxeterDiagramDisplay

`‣ CoxeterDiagramDisplay`( D ) | ( function )

`‣ CoxeterDiagramDisplay`( D, str ) | ( function )

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 str="mozilla".

This function requires Graphviz software.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7

##### 26.1-4 CoxeterDiagramFpArtinGroup

`‣ CoxeterDiagramFpArtinGroup`( D ) | ( function )

Inputs a Coxeter diagram D and returns the corresponding finitely presented Artin group.

Examples: 1

##### 26.1-5 CoxeterDiagramFpCoxeterGroup

`‣ CoxeterDiagramFpCoxeterGroup`( D ) | ( function )

Inputs a Coxeter diagram D and returns the corresponding finitely presented Coxeter group.

Examples: 1

##### 26.1-6 CoxeterDiagramIsSpherical

`‣ CoxeterDiagramIsSpherical`( D ) | ( function )

Inputs a Coxeter diagram D and returns "true" if the associated Coxeter groups is finite, and returns "false" otherwise.

Examples: 1

##### 26.1-7 CoxeterDiagramMatrix

`‣ CoxeterDiagramMatrix`( D ) | ( function )

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.

Examples:

##### 26.1-8 CoxeterSubDiagram

`‣ CoxeterSubDiagram`( D, V ) | ( function )

Inputs a Coxeter diagram D and a subset V of its vertices. It returns the full sub-diagram of D with vertex set V.

Examples:

##### 26.1-9 CoxeterDiagramVertices

`‣ CoxeterDiagramVertices`( D ) | ( function )

Inputs a Coxeter diagram D and returns its set of vertices.

Examples:

##### 26.1-10 EvenSubgroup

`‣ EvenSubgroup`( G ) | ( function )

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.

Examples: 1 , 2

##### 26.1-11 GraphOfGroupsDisplay

`‣ GraphOfGroupsDisplay`( D ) | ( function )

`‣ GraphOfGroupsDisplay`( D, str ) | ( function )

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 the second argument str="mozilla".

This function requires Graphviz software.

Examples: 1 , 2 , 3 , 4

##### 26.1-12 GraphOfResolutions

`‣ GraphOfResolutions`( D, n ) | ( function )

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.

Examples:

##### 26.1-13 GraphOfGroups

`‣ GraphOfGroups`( D ) | ( function )

Inputs a graph of resolutions D and returns the corresponding graph of groups.

Examples: 1 , 2 , 3 , 4

##### 26.1-14 GraphOfResolutionsDisplay

`‣ GraphOfResolutionsDisplay`( D ) | ( function )

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.

Examples:

##### 26.1-15 GraphOfGroupsTest

`‣ GraphOfGroupsTest`( D ) | ( function )

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.

Examples:

##### 26.1-16 TreeOfGroupsToContractibleGcomplex

`‣ TreeOfGroupsToContractibleGcomplex`( D, G ) | ( function )

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.

Examples:

##### 26.1-17 TreeOfResolutionsToContractibleGcomplex

`‣ TreeOfResolutionsToContractibleGcomplex`( D, G ) | ( function )

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.

Examples:

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