Difference between revisions of "ApCoCoA-1:Lamplighter group"
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=== <div id="Lamplighter_groups">[[:ApCoCoA:Symbolic data#Lamplighter_groups|Lamplighter groups]]</div> === | === <div id="Lamplighter_groups">[[:ApCoCoA:Symbolic data#Lamplighter_groups|Lamplighter groups]]</div> === | ||
==== Description ==== | ==== Description ==== | ||
| − | The standard presentation for the Lamplighter group arises from the wreath product structure | + | 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> | G = <a,b | (ab^{n}ab^{-n})^2 = 1> | ||
(Reference: Volodymyr Nekrashevych, 2005, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society) | (Reference: Volodymyr Nekrashevych, 2005, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society) | ||
Revision as of 14:12, 19 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, 2005, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society)
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);
