<html><head><title>[NQL] 1 Preface</title></head>
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<h1>1 Preface</h1><p>
<p>
In 1980, Grigorchuk <a href="biblio.htm#Grigorchuk80"><cite>Grigorchuk80</cite></a> gave an example of an infinite,
finitely generated torsion group which provided a first explicit
counter-example to the General Burnside Problem. This counter-example
is nowadays called the <var>Grigorchuk group</var> and was originally defined
as a group of transformations of the unit interval which preserve the
Lebesgue measure. Beside being a counter-example to the General Burnside
Problem, the Grigorchuk group was a first example of a group with an
intermediate growth function (see <a href="biblio.htm#Grigorchuk83"><cite>Grigorchuk83</cite></a>) and was used in
the construction of a finitely presented amenable group which is not
elementary amenable (see <a href="biblio.htm#Grigorchuk98"><cite>Grigorchuk98</cite></a>).
<p>
The Grigorchuk group is not finitely presentable
(see <a href="biblio.htm#Grigorchuk99"><cite>Grigorchuk99</cite></a>). However, in 1985, Igor Lysenok
(see <a href="biblio.htm#Lysenok85"><cite>Lysenok85</cite></a>) determined the following recursive presentation
for the Grigorchuk group:
<p><var> langlea,b,c,dmida<sup>2</sup>,b<sup>2</sup>,c<sup>2</sup>,d<sup>2</sup>,bcd,[d,d<sup>a</sup>]<sup>sigma^n</sup>,[d,d<sup>acaca</sup>]
<sup>sigma^n</sup>, (nin<font face="helvetica,arial">N</font>)rangle,<p></var>
where <var>sigma</var> is the homomorphism of the free group over <var>{a,b,c,d}</var>
which is induced by <var>amapstoc<sup>a</sup>, bmapstod, cmapstob</var>, and
<var>dmapstoc</var>. Hence, the infinitely many relators of this recursive
presentation can be described in finite terms using powers of the
endomorphism <var>sigma</var>.
<p>
In 2003, Bartholdi <a href="biblio.htm#Bartholdi03"><cite>Bartholdi03</cite></a> introduced the notion of an
<var><var>L</var>-presentation</var> for presentations of this type; that is, a group
presentation of the form
<p><var> G=leftlangleS left| Qcupbigcup<sub>varphiinPhi^*</sub>
R^varphiright.rightrangle,<p></var>
where <var>Phi^*</var> denotes the free monoid generated by a set of free group
endomorphisms <var>Phi</var>. He proved that various branch groups are finitely
<var>L</var>-presented but not finitely presentable and that every free group
in a variety of groups satisfying finitely many identities is finitely
<var>L</var>-presented (e.g. the Free Burnside- and the Free <var>n</var>-Engel groups).
<p>
The <font face="Gill Sans,Helvetica,Arial">NQL</font>-package defines new <font face="Gill Sans,Helvetica,Arial">GAP</font> objects to work with finitely
<var>L</var>-presented groups. The main part of the package is a nilpotent quotient
algorithm for finitely <var>L</var>-presented groups; that is, an algorithm which
takes as input a finitely<var>L</var>-presented group <var>G</var> and a positive integer
<var>c</var>. It computes a polycyclic presentation for the lower central series
quotient <var>G/gamma<sub>c+1</sub>(G)</var>. Therefore, a nilpotent quotient algorithm
can be used to determine the abelian invariants of the lower central
series sections <var>gamma<sub>c</sub>(G)/gamma<sub>c+1</sub>(G)</var> and the largest nilpotent
quotient of <var>G</var> if it exists.
<p>
Our nilpotent quotient algorithm generalizes Nickel's algorithm for
finitely presented groups (see <a href="biblio.htm#Nickel96"><cite>Nickel96</cite></a>) which is implemented in
the <font face="Gill Sans,Helvetica,Arial">NQ</font>-package; see <a href="biblio.htm#nq"><cite>nq</cite></a>. In difference to the <font face="Gill Sans,Helvetica,Arial">NQ</font>-package,
the <font face="Gill Sans,Helvetica,Arial">NQL</font>-package is implemented in <font face="Gill Sans,Helvetica,Arial">GAP</font> only.
<p>
Since finite <var>L</var>-presentations generalize finite presentations, our
algorithm also applies to finitely presented groups. It coincides with
Nickel's algorithm in this special case.
<p>
Our algorithm can be readily modified to determine the <var>p</var>-quotients of
a finitely <var>L</var>-presented group. An implementation is planned for future
expansions of the package.
<p>
A detailed description of our algorithm can be found in <a href="biblio.htm#BEH07"><cite>BEH07</cite></a>
or in the diploma thesis <a href="biblio.htm#H08"><cite>H08</cite></a> which is publicly available from
the website <a href="http://www-public.tu-bs.de:8080/~y0019492/pub/index.html">http://www-public.tu-bs.de:8080/~y0019492/pub/index.html</a>
<p>
<p>
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<P>
<address>NQL manual<br>November 2008
</address></body></html>
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