ApCoCoA-1:Thompson group
From ApCoCoAWiki
Revision as of 10:51, 28 March 2014 by StrohmeierB (talk | contribs) (→Example in Symbolic Data Format)
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;
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*c*d*a*a-1</ncpoly> </basis> <Comment>The partial LLex Gb has 249 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*c*d*a*a </Comment>