# 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 relation of the invers 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);
```