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, 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);