Difference between revisions of "ApCoCoA-1:Lamplighter group"
From ApCoCoAWiki
| Line 16: | Line 16: | ||
Use ZZ/(2)[a,b,c,d]; | Use ZZ/(2)[a,b,c,d]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsLamplighter() | Define CreateRelationsLamplighter() | ||
Relations:=[]; | Relations:=[]; | ||
| + | |||
// add the relation of the invers elements | // add the relation of the invers elements | ||
Append(Relations,[[a,c],[1]]); | Append(Relations,[[a,c],[1]]); | ||
| Line 45: | Line 47: | ||
Append(Relations, [RelationBuffer,[1]]); | Append(Relations, [RelationBuffer,[1]]); | ||
EndFor; | EndFor; | ||
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
| Line 50: | Line 53: | ||
Relations:=CreateRelationsLamplighter(); | Relations:=CreateRelationsLamplighter(); | ||
Relations; | Relations; | ||
| − | + | ||
| − | Size( | + | Gb:=NC.GB(Relations,31,1,100,1000); |
| + | Size(Gb); | ||
Revision as of 10:53, 23 August 2013
Description
The standard presentation for the Lamplighter group arises from the wreath product structure and can be simplified:
G = <a,b | (ab^{n}ab^{-n})^2 = 1>
Reference
Volodymyr Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society, 2005.
Computation
/*Use the ApCoCoA package ncpoly.*/
// Boundary of Lamplighter group
MEMORY.N:=3;
// a invers to c, b invers to d
Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");
Define CreateRelationsLamplighter()
Relations:=[];
// add the relation of the invers elements
Append(Relations,[[a,c],[1]]);
Append(Relations,[[c,a],[1]]);
Append(Relations,[[b,d],[1]]);
Append(Relations,[[d,b],[1]]);
// add the relation (ab^{n}ab^{-n})^2 = 1
For Index0 := 1 To MEMORY.N Do
RelationBuffer:=[];
Append(RelationBuffer,a);
For Index1 := 1 To Index0 Do
Append(RelationBuffer,b);
EndFor;
Append(RelationBuffer,a);
For Index1 := 1 To Index0 Do
Append(RelationBuffer,d);
EndFor;
Append(RelationBuffer,a);
For Index1 := 1 To Index0 Do
Append(RelationBuffer,b);
EndFor;
Append(RelationBuffer,a);
For Index1 := 1 To Index0 Do
Append(RelationBuffer,d);
EndFor;
Append(Relations, [RelationBuffer,[1]]);
EndFor;
Return Relations;
EndDefine;
Relations:=CreateRelationsLamplighter();
Relations;
Gb:=NC.GB(Relations,31,1,100,1000);
Size(Gb);
