GAP (HAP commands) - Chapter 17: Generators and relators of groups

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17 Generators and relators of groups
17.1
17.1-1 CayleyGraphOfGroupDisplay
17.1-2 IdentityAmongRelatorsDisplay
17.1-3 IsAspherical
17.1-4 PresentationOfResolution
17.1-5 TorsionGeneratorsAbelianGroup

### 17 Generators and relators of groups

#### 17.1

##### 17.1-1 CayleyGraphOfGroupDisplay

`‣ CayleyGraphOfGroupDisplay`( G, X ) | ( function )

`‣ CayleyGraphOfGroupDisplay`( G, X, str ) | ( function )

Inputs a finite group \(G\) together with a subset \(X\) of \(G\). 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 \(str\)="mozilla".

The argument \(G\) can also be a finite set of elements in a (possibly infinite) group containing \(X\). The edges of the graph are coloured according to which element of \(X\) they are labelled by. The list \(X\) corresponds to the list of colours [blue, red, green, yellow, brown, black] in that order.

This function requires Graphviz software.

Examples:

##### 17.1-2 IdentityAmongRelatorsDisplay

`‣ IdentityAmongRelatorsDisplay`( R, n ) | ( function )

`‣ IdentityAmongRelatorsDisplay`( R, n, str ) | ( function )

Inputs a free \(ZG\)-resolution \(R\) and an integer \(n\). 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 \(str\)="mozilla". (The resolution \(R\) should be reduced and, preferably, in dimension 1 it should correspond to a Cayley graph for \(G\). )

This function uses GraphViz software.

Examples: 1 , 2

##### 17.1-3 IsAspherical

`‣ IsAspherical`( F, R ) | ( function )

Inputs a free group \(F\) and a set \(R\) of words in \(F\). It performs a test on the 2-dimensional CW-space \(K\) associated to this presentation for the group \(G=F/\)<\(R\)>\(^F\).

The function returns "true" if \(K\) has trivial second homotopy group. In this case it prints: Presentation is aspherical.

Otherwise it returns "fail" and prints: Presentation is NOT piece-wise Euclidean non-positively curved. (In this case \(K\) 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.)

The function uses Polymake software.

Examples: 1 , 2 , 3 , 4

##### 17.1-4 PresentationOfResolution

`‣ PresentationOfResolution`( R ) | ( function )

Inputs at least two terms of a reduced \(ZG\)-resolution \(R\) and returns a record \(P\) with components

- \(P.freeGroup\) is a free group \(F\),

- \(P.relators\) is a list \(S\) of words in \(F\),

- \(P.gens\) is a list of positive integers such that the \(i\)-th generator of the presentation corresponds to the group element R!.elts[P[i]] .

where \(G\) is isomorphic to \(F\) modulo the normal closure of \(S\). This presentation for \(G\) corresponds to the 2-skeleton of the classifying CW-space from which \(R\) was constructed. The resolution \(R\) requires no contracting homotopy.

Examples: 1 , 2 , 3

##### 17.1-5 TorsionGeneratorsAbelianGroup

`‣ TorsionGeneratorsAbelianGroup`( G ) | ( function )

Inputs an abelian group \(G\) and returns a generating set \([x_1, \ldots ,x_n]\) where no pair of generators have coprime orders.

Examples:

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