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# 1 Preface

In 1980, Grigorchuk 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 Grigor80 also showed that this group is not finitely presented. In 1985, Lysenok Lysenok85 determined the following presentation for the Grigorchuk group

⟨a,b,c,d| a2,b2,c2,d2,bcd,[d,da]σn, [d,dacaca]σn, (n∈N) ⟩,
where σ is the homomorphism of the free group on a,b,c,d induced by a→ ca, b→d, c→b, and d→c. Thus the infinitely many relators of this presentation can be described in finite terms using powers of the endomorphism σ.

In 2003, Bartholdi 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 G/γc+1(G).

The nilpotent quotient algorithm defined in this package generalizes the method by Nickel Nickel96 as implemented in the NQ-package of GAP: see 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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