ApCoCoA:Baumslag groups

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Description

Baumslag-Solitar groups have the following presentation.

BS(m,n)<a, b | ba^{m} = a^{n}b> where m, n are natural numbers

Reference

G. Baumslag and D. Solitar, Some two generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. , 689 (1962) pp. 199–201.

Computation

We enumerate partial Groebner bases for the Baumslag-Solitar groups as follows.

/*Use the ApCoCoA package ncpoly.*/
Use ZZ/(2)[a[1..2],b[1..2]];
NC.SetOrdering("LLEX");
Define CreateRelations()
  A1:=[[a[1],a[2]],[1]];
  A2:=[[a[2],a[1]],[1]];
  B1:=[[b[1],b[2]],[1]];
  B2:=[[b[2],b[1]],[1]];
  R:=[[b[1],a[1]^(2)],[a[1]^(3),b[1]]];
  Return [A1,A2,B1,B2,R];
EndDefine;
-- Relation ba^2=a^3b. Change 2 and 3 in "()" to make another relation
G:=CreateRelations();
-- Enumerate a partial Groebner basis (see NC.GB for more details)
NC.GB(G,31,1,100,1000);
/*Use the ApCoCoA package gbmr.*/
-- See NCo.BGB for more details on the parameters DB, LB and OFlag.
Define BS(M,N,DB,LB,OFlag)
  $apcocoa/gbmr.SetX("aAbB");
  $apcocoa/gbmr.SetOrdering("LLEX");
  G:= [["aA",""],["Aa",""],["bB",""],["bB",""]];
  BA:= "b";
  AB:= "b";
  For I:= 1 To ARGV[1] Do
    BA:= BA + "a"; 
  EndFor;
  For I:= 1 To ARGV[2] Do
    AB:= "a" + Ab; 
  EndFor;
  Append(G,[BA,AB]);
  Return $apcocoa/gbmr.BGB(G,DB,LB,OFlag);
EndDefine;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-03-12" createdBy="strohmeier">
 	<vars>a1,a2,b1,b2</vars>
 	<uptoDeg>9</uptoDeg>
 	<basis>
 	<ncpoly>a1*a2-1</ncpoly>
 	<ncpoly>a2*a1-1</ncpoly>
 	<ncpoly>b1*b2-1</ncpoly>
 	<ncpoly>b2*b1-1</ncpoly>
 	<ncpoly>b1*a1^2-a1^3*b1</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 208 elements</Comment>
 	<Comment>Baumslag-Solitar_group1</Comment>
 </FREEALGEBRA>