ApCoCoA-1:VonDyck groups

Von Dyck groups

Description

The von Dyck groups are sometimes referred to as ordinary triangle groups and are subgroups of index 2 in Triangle(l, m, n) generated by words of even length in the generators a, b, c. A specific representation is given for x = ab, y = ca, yx = cb:

 D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>

(Reference: not found yet)

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Parameters of von Dyck group
 MEMORY.L:=3;
 MEMORY.M:=5;
 MEMORY.N:=2;
 
 Use ZZ/(2)[x,y];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsVonDyck()
   Relations:=[];
   
   // add the relation x^l = 1
   Append(Relations,[[x^MEMORY.L],[1]]);
   
   // add the relation y^m = 1
   Append(Relations,[[y^MEMORY.M],[1]]);
   
   // add the relation (xy)^n = 1
   BufferXY:=[];
   For Index1 := 1 To MEMORY.N Do
   	Append(BufferXY,x);
   	Append(BufferXY,y);
   EndFor;
   Append(Relations,[BufferXY,[1]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsVonDyck();
 Relations;
 
 Gb:=NC.GB(Relations);
 Gb;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
 	<vars>x,y</vars>
 	<basis>
 	<ncpoly>x^3-1</ncpoly>
 	<ncpoly>y^5-1</ncpoly>
 	<ncpoly>(x*y)^2-1</ncpoly>
 	</basis>
 	<Comment>Von_Dyck_group_l3m5n2</Comment>
 </FREEALGEBRA>