ApCoCoA-1:Baumslag-Gersten groups

Baumslag-Gersten Groups

Description

The Baumslag-Gersten groups have a Dehn function growing faster than any fixed iterated tower of exponentials and can be represented as:

 BG = <a,b | (b^{-1}a^{-1}b)a(b^{-1}a^{-1}b) = a^{2}>

Reference

A. N. Platonov, An isoparametric function of the Baumslag-Gersten group. (in Russian.) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2004, , no. 3, pp. 12–17; translation in: Moscow University Mathematics Bulletin, vol. 59 (2004), no. 3, pp. 12–17 (2005).

Computation

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

/*Use the ApCoCoA package ncpoly.*/
 
 Use ZZ/(2)[a,b,c,d];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsBaumslagGersten()
   Relations:=[];
 
   // Add the relations of the inverse elements ac = ca = 1 and bd = db = 1
   Append(Relations,[[a,c],[1]]);
   Append(Relations,[[c,a],[1]]);
   Append(Relations,[[b,d],[1]]);
   Append(Relations,[[d,b],[1]]);
 
   // Add the relation (b^{-1}a^{-1}b)a(b^{-1}a^{-1}b = a^2)
   Append(Relations,[[d,c,b,a,d,c,b],[a^2]]);
 
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsBaumslagGersten();
 Relations;
 
   -- Enumerate a partial Groebner basis (see NC.GB for more details)
 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>d*c*b*a*d*c*b-a*a</ncpoly>
 	</basis>
 	<Comment>The partial LLex GB has 201 elements</Comment>
 	<Comment>Baumslag-Gersten_group</Comment>
 </FREEALGEBRA>