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11. Generators and relators of groups

### 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.
`IdentityAmongRelatorsDisplay(R,n)` `IdentityAmongRelatorsDisplay(R,n,"mozilla")` 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 a second argument. (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)` 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.

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