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

From ApCoCoAWiki
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
=== <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
Line 70: Line 70:
 
    
 
    
 
  // Define the variable k,n of the thompson group
 
  // Define the variable k,n of the thompson group
// MEMORY.N has to be MEMORY.K+2
 
 
  MEMORY.N:=5;
 
  MEMORY.N:=5;
MEMORY.K:=3;
 
 
   
 
   
 
  Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
 
  Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
Line 84: Line 82:
 
   EndFor;
 
   EndFor;
 
    
 
    
  For Index2 := 1 To MEMORY.K Do
 
 
   For Index1 := 2 To MEMORY.N-1 Do
 
   For Index1 := 2 To MEMORY.N-1 Do
 +
  For Index2 := 1 To MEMORY.N-2 Do
 
   If (Index1 > Index2) Then
 
   If (Index1 > Index2) Then
 
     Append(Relations,[[y[Index2],x[Index1],x[Index2]],[x[Index1+1]]]);
 
     Append(Relations,[[y[Index2],x[Index1],x[Index2]],[x[Index1+1]]]);
Line 99: Line 97:
 
  Gb:=NC.GB(Relations,31,1,100,1000);
 
  Gb:=NC.GB(Relations,31,1,100,1000);
 
  Gb;
 
  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>