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
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.
NQL manual