Difference between revisions of "ApCoCoA-1:Baumslag-Gersten groups"

From ApCoCoAWiki
Line 15: Line 15:
 
     Relations:=[];
 
     Relations:=[];
 
    
 
    
     // add the relations of the inverse elements ac = ca = 1 and bd = db = 1
+
     // Add the relations of the inverse elements ac = ca = 1 and bd = db = 1
 
     Append(Relations,[[a,c],[1]]);
 
     Append(Relations,[[a,c],[1]]);
 
     Append(Relations,[[c,a],[1]]);
 
     Append(Relations,[[c,a],[1]]);
Line 21: Line 21:
 
     Append(Relations,[[d,b],[1]]);
 
     Append(Relations,[[d,b],[1]]);
 
    
 
    
     // add the relation (b^{-1}a^{-1}b)a(b^{-1}a^{-1}b = a^2)
+
     // 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]]);
 
     Append(Relations,[[d,c,b,a,d,c,b],[a^2]]);
 
    
 
    
Line 30: Line 30:
 
   Relations;
 
   Relations;
 
    
 
    
   GB:=NC.GB(Relations,31,1,100,1000);
+
    -- Enumerate a partial Groebner basis (see NC.GB for more details)
   GB;
+
   Gb:=NC.GB(Relations,31,1,100,1000);
 +
   Gb;

Revision as of 07:20, 23 August 2013

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;