%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W intro.tex NQL Rene Hartung %% %H $Id: preface.tex,v 1.8 2008/08/28 07:48:06 gap Exp $ %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Preface} In 1980, Grigorchuk~\cite{Grigorchuk80} 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 <Grigorchuk group> 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 \cite{Grigorchuk83}) and was used in the construction of a finitely presented amenable group which is not elementary amenable (see~\cite{Grigorchuk98}). The Grigorchuk group is not finitely presentable (see~\cite{Grigorchuk99}). However, in 1985, Igor Lysenok (see~\cite{Lysenok85}) determined the following recursive presentation for the Grigorchuk group: $$ \langle a,b,c,d\mid a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n},[d,d^{acaca}] ^{\sigma^n}, (n\in\N)\rangle,$$ where $\sigma$ is the homomorphism of the free group over $\{a,b,c,d\}$ which is induced by $a\mapsto c^a, b\mapsto d, c\mapsto b$, and $d\mapsto c$. Hence, the infinitely many relators of this recursive presentation can be described in finite terms using powers of the endomorphism $\sigma$. In 2003, Bartholdi~\cite{Bartholdi03} introduced the notion of an <$L$-presentation> for presentations of this type; that is, a group presentation of the form $$ G=\left\langle S~\left|~ Q\cup \bigcup_{\varphi\in\Phi^\*} R^\varphi\right.\right\rangle,$$ where $\Phi^\*$ denotes the free monoid generated by a set of free group endomorphisms $\Phi$. He proved that various branch groups are finitely $L$-presented but not finitely presentable and that every free group in a variety of groups satisfying finitely many identities is finitely $L$-presented (e.g. the Free Burnside- and the Free $n$-Engel groups). The {\NQL}-package defines new {\GAP} objects to work with finitely $L$-presented groups. The main part of the package is a nilpotent quotient algorithm for finitely $L$-presented groups; that is, an algorithm which takes as input a finitely$L$-presented group $G$ and a positive integer $c$. It computes a polycyclic presentation for the lower central series quotient $G/\gamma_{c+1}(G)$. Therefore, a nilpotent quotient algorithm can be used to determine the abelian invariants of the lower central series sections $\gamma_c(G)/\gamma_{c+1}(G)$ and the largest nilpotent quotient of $G$ if it exists. Our nilpotent quotient algorithm generalizes Nickel's algorithm for finitely presented groups (see~\cite{Nickel96}) which is implemented in the {\NQ}-package; see~\cite{nq}. In difference to the {\NQ}-package, the {\NQL}-package is implemented in \GAP\ only. Since finite $L$-presentations generalize finite presentations, our algorithm also applies to finitely presented groups. It coincides with Nickel's algorithm in this special case. Our algorithm can be readily modified to determine the $p$-quotients of a finitely $L$-presented group. An implementation is planned for future expansions of the package. A detailed description of our algorithm can be found in~\cite{BEH07} or in the diploma thesis~\cite{H08} which is publicly available from the website \URL{http://www-public.tu-bs.de:8080/~y0019492/pub/index.html} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E preface.tex . . . . . . . . . . . . . . . . . . . . . . . . ends here