HAP home

11. Generators and relators of groups

CayleyGraphDisplay(G,X)
CayleyGraphDisplay(G,X,"mozilla")

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.

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.

IsAspherical(F,R)

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.

PresentationOfResolution(R)

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,

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.

TorsionGeneratorsAbelianGroup(G)

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


 





generated by GAPDoc2HTML