Difference between revisions of "ApCoCoA-1:Baumslag-Gersten groups"
From ApCoCoAWiki
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=== <div id="Baumslag-Gersten_groups">[[:ApCoCoA:Symbolic data#Baumslag_groups|Baumslag groups]]</div> === | === <div id="Baumslag-Gersten_groups">[[:ApCoCoA:Symbolic data#Baumslag_groups|Baumslag 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- | + | We enumerate partial Groebner bases for the Baumslag-Gersten groups as follows. |
/*Use the ApCoCoA package ncpoly.*/ | /*Use the ApCoCoA package ncpoly.*/ | ||
| Line 14: | Line 13: | ||
Define CreateRelationsBaumslagGersten() | Define CreateRelationsBaumslagGersten() | ||
Relations:=[]; | Relations:=[]; | ||
| − | // add the | + | // 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]]); | ||
Revision as of 07:07, 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;
GB:=NC.GB(Relations,31,1,100,1000);
GB;
