Difference between revisions of "ApCoCoA-1:Triangle groups"
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− | === <div id="Triangle groups">[[:ApCoCoA:Symbolic data#Triangle_groups|Triangle | + | === <div id="Triangle groups">[[:ApCoCoA:Symbolic data#Triangle_groups|Triangle Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
For integers l,m,n greater than 1 the Triangle groups have can be described with the following finite representation: | For integers l,m,n greater than 1 the Triangle groups have can be described with the following finite representation: |
Latest revision as of 21:05, 22 April 2014
Description
For integers l,m,n greater than 1 the Triangle groups have can be described with the following finite representation:
Triangle(l,m,n) = {a,b,c | a^{2} = b^{2} = c^{2} = (ab)^{l} = (bc)^{m} = (ca)^{n} = 1}
Reference
Gross, Jonathan L.; Tucker, Thomas W. (2001), "6.2.8 Triangle Groups", Topological graph theory, Courier Dover Publications
Computation
/*Use the ApCoCoA package ncpoly.*/ // set the variables l,m,n // Note that l,m,n have to be greater than 1 MEMORY.L:=3; MEMORY.M:=3; MEMORY.N:=3; Use ZZ/(2)[a,b,c]; NC.SetOrdering("LLEX"); Define CreateRelationsTriangle() Relations:=[]; // add the relation a^2 = b^2 = c^2 = 1 Append(Relations,[[a^2],[1]]); Append(Relations,[[b^2],[1]]); Append(Relations,[[c^2],[1]]); // add the relation (ab)^l = 1 RelationBuffer1:=[]; For Index0:= 1 To MEMORY.L Do Append(RelationBuffer1,a); Append(RelationBuffer1,b); EndFor; Append(Relations,[RelationBuffer1,[1]]); // add the relation (bc)^m = 1 RelationBuffer2:=[]; For Index1:= 1 To MEMORY.M Do Append(RelationBuffer2,b); Append(RelationBuffer2,c); EndFor; Append(Relations,[RelationBuffer2,[1]]); // add the relation (ca)^n = 1 RelationBuffer3:=[]; For Index2:= 1 To MEMORY.N Do Append(RelationBuffer3,c); Append(RelationBuffer3,a); EndFor; Append(Relations,[RelationBuffer3,[1]]); Return Relations; EndDefine; Relations:=CreateRelationsTriangle(); GB:=NC.GB(Relations);
Examples in Symbolic Data Format
Triangle group l=2 m=4 n=6
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a,b,c</vars> <basis> <ncpoly>a*a-1</ncpoly> <ncpoly>b*b*-1</ncpoly> <ncpoly>c*c*-1</ncpoly> <ncpoly>(a*b)^(2)-1</ncpoly> <ncpoly>(b*c)^(4)-1</ncpoly> <ncpoly>(c*a)^(6)-1</ncpoly> </basis> <Comment>Triangle_groups_l2m4n6</Comment> </FREEALGEBRA>
Triangle group l=3 m=3 n=3
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a,b,c</vars> <basis> <ncpoly>a*a-1</ncpoly> <ncpoly>b*b*-1</ncpoly> <ncpoly>c*c*-1</ncpoly> <ncpoly>(a*b)^(3)-1</ncpoly> <ncpoly>(b*c)^(3)-1</ncpoly> <ncpoly>(c*a)^(3)-1</ncpoly> </basis> <Comment>Triangle_groups_3</Comment> </FREEALGEBRA>
Triangle group l=3 m=4 n=5
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a,b,c</vars> <basis> <ncpoly>a*a-1</ncpoly> <ncpoly>b*b*-1</ncpoly> <ncpoly>c*c*-1</ncpoly> <ncpoly>(a*b)^(3)-1</ncpoly> <ncpoly>(b*c)^(4)-1</ncpoly> <ncpoly>(c*a)^(5)-1</ncpoly> </basis> <Comment>Triangle_groups_l3m4n5</Comment> </FREEALGEBRA>
Triangle group l=3 m=5 n=7
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a,b,c</vars> <basis> <ncpoly>a*a-1</ncpoly> <ncpoly>b*b*-1</ncpoly> <ncpoly>c*c*-1</ncpoly> <ncpoly>(a*b)^(3)-1</ncpoly> <ncpoly>(b*c)^(5)-1</ncpoly> <ncpoly>(c*a)^(7)-1</ncpoly> </basis> <Comment>Triangle_groups_l3m5n7</Comment> </FREEALGEBRA>
Triangle group l=7 m=11 n=13
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>a,b,c</vars> <basis> <ncpoly>a*a-1</ncpoly> <ncpoly>b*b*-1</ncpoly> <ncpoly>c*c*-1</ncpoly> <ncpoly>(a*b)^(7)-1</ncpoly> <ncpoly>(b*c)^(11)-1</ncpoly> <ncpoly>(c*a)^(13)-1</ncpoly> </basis> <Comment>Triangle_groups_l7m11n13</Comment> </FREEALGEBRA>