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],[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>