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, ... ,x_n] where no pair of generators have coprime orders.

Examples:

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