‣ 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:
‣ 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.
‣ 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.
‣ 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.
‣ 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:
generated by GAPDoc2HTML