Difference between revisions of "ApCoCoA-1:Lamplighter group"
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| − | === <div id=" | + | === <div id="Lamplighter_group">[[:ApCoCoA:Symbolic data#Lamplighter_group|Lamplighter Group]]</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> | |
| − | + | ||
| + | ==== Reference ==== | ||
| + | Volodymyr Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society, 2005. | ||
==== Computation ==== | ==== Computation ==== | ||
| Line 14: | 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 | + | |
| + | // add the relations of the inverse elements | ||
Append(Relations,[[a,c],[1]]); | Append(Relations,[[a,c],[1]]); | ||
Append(Relations,[[c,a],[1]]); | Append(Relations,[[c,a],[1]]); | ||
| Line 43: | Line 47: | ||
Append(Relations, [RelationBuffer,[1]]); | Append(Relations, [RelationBuffer,[1]]); | ||
EndFor; | EndFor; | ||
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
| Line 48: | Line 53: | ||
Relations:=CreateRelationsLamplighter(); | Relations:=CreateRelationsLamplighter(); | ||
Relations; | Relations; | ||
| − | + | ||
| − | Size( | + | Gb:=NC.GB(Relations,31,1,100,1000); |
| + | Size(Gb); | ||
| + | ====Example in Symbolic Data Format==== | ||
| + | <FREEALGEBRA createdAt="2014-02-27" createdBy="strohmeier"> | ||
| + | <vars>a,b,c,d</vars> | ||
| + | <uptoDeg>13</uptoDeg> | ||
| + | <basis> | ||
| + | <ncpoly>a*c-1</ncpoly> | ||
| + | <ncpoly>c*a-1</ncpoly> | ||
| + | <ncpoly>b*d-1</ncpoly> | ||
| + | <ncpoly>d*b-1</ncpoly> | ||
| + | <ncpoly>(a*(b^1)*a*(d^1))^2-1</ncpoly> | ||
| + | <ncpoly>(a*(b^2)*a*(d^2))^2-1</ncpoly> | ||
| + | <ncpoly>(a*(b^3)*a*(d^3))^2-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>The partial LLex Gb has 191 elements</Comment> | ||
| + | <Comment>Lamplighter_group_3</Comment> | ||
| + | </FREEALGEBRA> | ||
Latest revision as of 20:58, 22 April 2014
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 relations of the inverse 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);
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-02-27" createdBy="strohmeier"> <vars>a,b,c,d</vars> <uptoDeg>13</uptoDeg> <basis> <ncpoly>a*c-1</ncpoly> <ncpoly>c*a-1</ncpoly> <ncpoly>b*d-1</ncpoly> <ncpoly>d*b-1</ncpoly> <ncpoly>(a*(b^1)*a*(d^1))^2-1</ncpoly> <ncpoly>(a*(b^2)*a*(d^2))^2-1</ncpoly> <ncpoly>(a*(b^3)*a*(d^3))^2-1</ncpoly> </basis> <Comment>The partial LLex Gb has 191 elements</Comment> <Comment>Lamplighter_group_3</Comment> </FREEALGEBRA>
