# Difference between revisions of "ApCoCoA-1:Lamplighter group"

### Lamplighter group

#### 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],]);
Append(Relations,[[c,a],]);
Append(Relations,[[b,d],]);
Append(Relations,[[d,b],]);

// 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,]);
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 LLexGb has 191 elements</Comment>
<Comment>Lamplighter_group_3</Comment>
</FREEALGEBRA>
```