Difference between revisions of "ApCoCoA-1:Thompson group"

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=== <div id="Thompson_groups">[[:ApCoCoA:Symbolic data#Thompson_group|Thompson group]]</div> ===
+
=== <div id="Thompson_groups">[[:ApCoCoA:Symbolic data#Thompson_group|Thompson Group]]</div> ===
 
==== Description ====
 
==== 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
 
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:
 
two. A representation is given by:
 
   T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1>
 
   T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1>
 +
or alternative:
 +
  Th = <x_{0},x_{1},x_{2},... | x_{k}^{-1}x_{n}x_{k} = x_{n+1} for all k < n>
  
 
==== Reference ====
 
==== Reference ====
Line 9: Line 11:
  
 
==== Computation ====
 
==== Computation ====
/*Use the ApCoCoA package ncpoly.*/
+
/*Use the ApCoCoA package ncpoly.*/
 
   
 
   
 
  Use ZZ/(2)[a,b,c,d];
 
  Use ZZ/(2)[a,b,c,d];
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   //add the relation [ad,a^{-1}ba] = 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
+
   // the commutator of [ad,a^{-2}ba^2] is a,d,c,c,b,a,a,b,c,c^2,d,a^2
   Append(Relations,[[a,d,c,c,b,a,a,b,c,c,c,d,a,a],[1]]);
+
   Append(Relations,[[a,d,c,c,b,a,a,b,c,c^2,d,a^2],[1]]);
 
    
 
    
 
   Return Relations;
 
   Return Relations;
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   [ad,ccbaa]=a*d*c*c*b*a*a*b*c*c^2*d*a^2
 
   [ad,ccbaa]=a*d*c*c*b*a*a*b*c*c^2*d*a^2
 
   </Comment>
 
   </Comment>
 +
 +
==== Alternative Computation ====
 +
 +
/*Use the ApCoCoA package ncpoly.*/
 +
 
 +
// Define the variable k,n of the thompson group
 +
MEMORY.N:=5;
 +
 +
Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
 +
NC.SetOrdering("LLEX");
 +
 +
Define CreateRelationsthomp()
 +
  Relations:=[];
 +
  For Index1 := 1 To MEMORY.N Do
 +
    Append(Relations,[[x[Index1],y[Index1]],[1]]);
 +
    Append(Relations,[[y[Index1],x[Index1]],[1]]);
 +
  EndFor;
 +
 
 +
  For Index1 := 2 To MEMORY.N-1 Do
 +
  For Index2 := 1 To MEMORY.N-2 Do
 +
  If (Index1 > Index2) Then
 +
    Append(Relations,[[y[Index2],x[Index1],x[Index2]],[x[Index1+1]]]);
 +
  EndIf 
 +
  EndFor;
 +
  EndFor;
 +
  Return Relations;
 +
EndDefine;
 +
 +
Relations:=CreateRelationsthomp();
 +
Relations;
 +
 +
Gb:=NC.GB(Relations,31,1,100,1000);
 +
Gb;
 +
 +
====Example in Symbolic Data Format====
 +
  <FREEALGEBRA createdAt="2014-03-28" createdBy="strohmeier">
 +
  <vars>x1,x2,x3,x4,x5,y1,y2,y3,y4,y5</vars>
 +
  <uptoDeg>4</uptoDeg>
 +
  <basis>
 +
  <ncpoly>x1*y1-1</ncpoly>
 +
  <ncpoly>y1*x1-1</ncpoly>
 +
  <ncpoly>x2*y2-1</ncpoly>
 +
  <ncpoly>y2*x2-1</ncpoly>
 +
  <ncpoly>x3*y3-1</ncpoly>
 +
  <ncpoly>y3*x3-1</ncpoly>
 +
  <ncpoly>x4*y4-1</ncpoly>
 +
  <ncpoly>y4*x4-1</ncpoly>
 +
  <ncpoly>x5*y5-1</ncpoly>
 +
  <ncpoly>y5*x5-1</ncpoly>
 +
  <ncpoly>y1*x2*x1-x3</ncpoly>
 +
  <ncpoly>y1*x3*x1-x4</ncpoly>
 +
  <ncpoly>y1*x4*x1-x5</ncpoly>
 +
  <ncpoly>y2*x3*x2-x4</ncpoly>
 +
  <ncpoly>y2*x4*x2-x5</ncpoly>
 +
  <ncpoly>y3*x4*x3-x5</ncpoly>
 +
  </basis>
 +
  <Comment>The partial LLex Gb has 126 elements</Comment>
 +
  <Comment>Thompson_group_alt5</Comment>
 +
  </FREEALGEBRA>

Latest revision as of 21:05, 22 April 2014

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>

or alternative:

  Th = <x_{0},x_{1},x_{2},... | x_{k}^{-1}x_{n}x_{k} = x_{n+1} for all k < n>

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^2,d,a^2
  Append(Relations,[[a,d,c,c,b,a,a,b,c,c^2,d,a^2],[1]]);
  
  Return Relations;
EndDefine;

Relations:=CreateRelationsThompson();
Relations;

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

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>11</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>a*d*c*b*a*b*c*c*d*a-1</ncpoly>
 	<ncpoly>a*d*c*c*b*a*a*b*c*c^2*d*a^2-1</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 393 elements</Comment>
 	<Comment>Thompson_group</Comment>
 </FREEALGEBRA>
 
 
 <Comment> Commutators
 [g,h] = ghg^{-1}h^{-1}
 [ad,cba]=a*d*c*b*a*b*c*c*d*a
 [ad,ccbaa]=a*d*c*c*b*a*a*b*c*c^2*d*a^2
 </Comment>

Alternative Computation

/*Use the ApCoCoA package ncpoly.*/
 
// Define the variable k,n of the thompson group
MEMORY.N:=5;

Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
NC.SetOrdering("LLEX");

Define CreateRelationsthomp()
  Relations:=[];
  For Index1 := 1 To MEMORY.N Do
    Append(Relations,[[x[Index1],y[Index1]],[1]]);
    Append(Relations,[[y[Index1],x[Index1]],[1]]);
  EndFor;
  
  For Index1 := 2 To MEMORY.N-1 Do
  For Index2 := 1 To MEMORY.N-2 Do
  If (Index1 > Index2) Then
    Append(Relations,[[y[Index2],x[Index1],x[Index2]],[x[Index1+1]]]);
  EndIf  
  EndFor;
  EndFor;
  Return Relations;
EndDefine;

Relations:=CreateRelationsthomp();
Relations;

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

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-03-28" createdBy="strohmeier">
 	<vars>x1,x2,x3,x4,x5,y1,y2,y3,y4,y5</vars>
 	<uptoDeg>4</uptoDeg>
 	<basis>
 	<ncpoly>x1*y1-1</ncpoly>
 	<ncpoly>y1*x1-1</ncpoly>
 	<ncpoly>x2*y2-1</ncpoly>
 	<ncpoly>y2*x2-1</ncpoly>
 	<ncpoly>x3*y3-1</ncpoly>
 	<ncpoly>y3*x3-1</ncpoly>
 	<ncpoly>x4*y4-1</ncpoly>
 	<ncpoly>y4*x4-1</ncpoly>
 	<ncpoly>x5*y5-1</ncpoly>
 	<ncpoly>y5*x5-1</ncpoly>
 	<ncpoly>y1*x2*x1-x3</ncpoly>
 	<ncpoly>y1*x3*x1-x4</ncpoly>
 	<ncpoly>y1*x4*x1-x5</ncpoly>
 	<ncpoly>y2*x3*x2-x4</ncpoly>
 	<ncpoly>y2*x4*x2-x5</ncpoly>
 	<ncpoly>y3*x4*x3-x5</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 126 elements</Comment>
 	<Comment>Thompson_group_alt5</Comment>
 </FREEALGEBRA>