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

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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 ==== |

## Revision as of 12:28, 28 March 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>