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%W  intro.tex			NQL				Rene Hartung
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%H  $Id: preface.tex,v 1.7 2007/05/25 08:30:53 gap Exp $
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\Chapter{Preface}

In 1980, Grigorchuk \cite{Grigor80} gave an example of an infinite, finitely
generated torsion group, which provided a counter-example for the general
Burnside problem. It is nowadays called the Grigorchuk group and was
originally defined
as a group of transformations of the interval $[0,1]$ which preserve the
Lebesgue measure. Grigorchuk \cite{Grigor80} also showed that this group
is not finitely presented. In 1985, Lysenok \cite{Lysenok85} determined the 
following presentation for the Grigorchuk group
%display{nonhtml}
$$ \langle a,b,c,d\mid a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n},[d,d^{acaca}]
   ^{\sigma^n}, (n\in\N)\rangle,$$
%enddisplay
%display{html}
%<center>&lang;<var>a,b,c,d</var>| <var>a<sup>2</sup>,b<sup>2</sup>,c<sup>2</sup>,d<sup>2</sup>,bcd,[d,d<sup>a</sup>]<sup>&sigma;<sup>n</sup></sup>, [d,d<sup>acaca</sup>]<sup>&sigma;<sup>n</sup></sup>, (n&isin;N)</var> &rang;,</center>
%enddisplay
where 
%display{nonhtml}
$\sigma$
%enddisplay
%display{html}
%&sigma;
%enddisplay
is the homomorphism of the free group on $a,b,c,d$ induced by 
%display{nonhtml}
$a\mapsto c^a, b\mapsto d, c\mapsto b$, and $d\mapsto c$.
%enddisplay
%display{html}
%<var>a&rarr; c<sup>a</sup>, b&rarr;d, c&rarr;b, </var> and <var> d&rarr;c.</var> 
%enddisplay
Thus the infinitely many relators of this presentation can be described in finite 
terms using powers of the endomorphism 
%display{nonhtml}
$\sigma$.
%enddisplay
%display{html}
%&sigma;.
%enddisplay

In 2003, Bartholdi \cite{Bartholdi03} introduced the notion of an L-presented
group for groups of this type (see Chapter~"Introduction to L-presented
groups" for a precise definition of L-presented groups).  He proved that
each finitely generated, contracting, semi-fractal, regular branch group is
finitely L-presented but not finitely presented. 

The {\NQL}-package defines new {\GAP} objects to work with
L-presented groups. The main part of the package is a nilpotent 
quotient algorithm for L-presented groups. That is an algorithm
which takes as input an L-presented group $G$ and a positive integer $c$. 
It computes a polycyclic presentation for the lower central series quotient
%display{nonhtml}
$G/\gamma_{c+1}(G)$.
%enddisplay
%display{html}
%<var>G/&gamma;<sub>c+1</sub>(G)</var>.
%enddisplay

The nilpotent quotient algorithm defined in this package generalizes the method
by Nickel \cite{Nickel96} as implemented in the {\NQ}-package of GAP: see
\cite{nq}. In difference to {\NQ}, the {\NQL}-package is implemented in GAP.

Our method can be readily modified to determine $p$-quotients of a finitely
L-presented group. An implementation of a PQ is planned for future expansions
of the {\NQL}-package.

As finite presentations can be considered as a special type of finite
L-presentations, our algorithm also applies to finitely presented groups.
It coincides with Nickel's {\NQ} in this special case. 

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