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

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(New page: === <div id="Baumslag-Gersten_groups">Baumslag groups</div> === ==== Description ==== The Baumslag-Gersten groups have a Dehn function growing fa...)
 
 
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=== <div id="Baumslag-Gersten_groups">[[:ApCoCoA:Symbolic data#Baumslag_groups|Baumslag groups]]</div> ===
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=== <div id="Baumslag-Gersten_groups">[[:ApCoCoA:Symbolic data#Baumslag_groups|Baumslag-Gersten Groups]]</div> ===
 
==== Description ====
 
==== Description ====
The Baumslag-Gersten groups have a Dehn function growing faster than any fixed iterated tower of exponentials and can
+
The Baumslag-Gersten groups have a Dehn function growing faster than any fixed iterated tower of exponentials and can be represented as:
be represented as:
 
 
   BG = <a,b | (b^{-1}a^{-1}b)a(b^{-1}a^{-1}b) = a^{2}>
 
   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).)
+
 
 +
==== 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 ====
 
==== Computation ====
We enumerate partial Groebner bases for the Baumslag-Solitar groups as follows.
+
We enumerate partial Groebner bases for the Baumslag-Gersten groups as follows.
 
  /*Use the ApCoCoA package ncpoly.*/
 
  /*Use the ApCoCoA package ncpoly.*/
 
    
 
    
 
   Use ZZ/(2)[a,b,c,d];
 
   Use ZZ/(2)[a,b,c,d];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
 +
 
 
   Define CreateRelationsBaumslagGersten()
 
   Define CreateRelationsBaumslagGersten()
 
     Relations:=[];
 
     Relations:=[];
     // add the relation of the invers elements ac = ca = 1 and bd = db = 1
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 +
     // 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]]);
 
     Append(Relations,[[b,d],[1]]);
 
     Append(Relations,[[b,d],[1]]);
 
     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)
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     // 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]]);
 +
 
 
     Return Relations;
 
     Return Relations;
 
   EndDefine;
 
   EndDefine;
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   Relations:=CreateRelationsBaumslagGersten();
 
   Relations:=CreateRelationsBaumslagGersten();
 
   Relations;
 
   Relations;
   GB:=NC.GB(Relations,31,1,100,1000);
+
    
   GB;
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    -- 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>

Latest revision as of 12:18, 19 April 2014

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>