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 | + | === <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). | ||
==== 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.*/ | ||
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 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) |
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; | ||
| Line 27: | Line 31: | ||
Relations:=CreateRelationsBaumslagGersten(); | Relations:=CreateRelationsBaumslagGersten(); | ||
Relations; | Relations; | ||
| − | GB:=NC.GB(Relations,31,1,100,1000); | + | |
| − | GB | + | -- 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>
