ApCoCoA-1:Thompson group

Thompson group

Description

The Thompson group can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the unit interval which have breakpoints only at dyadic points and on intervals of differentiability the slopes are powers of two. A representation is given by:

  T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1>

Reference

NEW PRESENTATIONS OF THOMPSON'S GROUPS AND APPLICATIONS: UFFE HAAGERUP AND GABRIEL PICIOROAGA

Computation

/*Use the ApCoCoA package ncpoly.*/

Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");

 Define CreateRelationsThompson()
  Relations:=[];
  
   // add the inverse relations
  Append(Relations,[[a,c],[1]]);
  Append(Relations,[[c,a],[1]]);
  Append(Relations,[[b,d],[1]]);
  Append(Relations,[[d,b],[1]]);
 
  //add the relation [ad,a^{-1}ba] = 1
  // the commutator of [ad,a^{-1}ba] is a,d,c,b,a,b,c,c,d,a
  Append(Relations,[[a,d,c,b,a,b,c,c,d,a],[1]]);
 
  //add the relation [ad,a^{-1}ba] = 1
  // the commutator of [ad,a^{-2}ba^2] is a,d,c,c,b,a,a,b,c,c,c,d,a,a
  Append(Relations,[[a,d,c,c,b,a,a,b,c,c,c,d,a,a],[1]]);
  
  Return Relations;
EndDefine;

Relations:=CreateRelationsThompson();
Relations;

Gb:=NC.GB(Relations,31,1,100,1000);
Gb;