<p>This package was written by René Hartung in 2009; maintenance has been overtaken by Laurent Bartholdi, who translated the manual to <strong class="pkg">GAPDoc</strong>.</p>
<p>In 1980, Grigorchuk <a href="chapBib_mj.html#biBGrigorchuk80">[Gri80]</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 <em>Grigorchuk group</em> 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="chapBib_mj.html#biBGrigorchuk83">[Gri83]</a>) and was used in the construction of a finitely presented amenable group which is not elementary amenable (see <a href="chapBib_mj.html#biBGrigorchuk98">[Gri98]</a>).</p>
<p>The Grigorchuk group is not finitely presentable (see <a href="chapBib_mj.html#biBGrigorchuk99">[Gri99]</a>). However, in 1985, Igor Lysenok (see <a href="chapBib_mj.html#biBLysenok85">[Lys85]</a>) determined the following recursive presentation for the Grigorchuk group:</p>
<p>where <span class="SimpleMath">\(\sigma\)</span> is the homomorphism of the free group over <span class="SimpleMath">\(\{a,b,c,d\}\)</span> which is induced by <span class="SimpleMath">\(a\mapsto c^a, b\mapsto d, c\mapsto b\)</span>, and <span class="SimpleMath">\(d\mapsto c\)</span>. Hence, the infinitely many relators of this recursive presentation can be described in finite terms using powers of the endomorphism <span class="SimpleMath">\(\sigma\)</span>.</p>
<p>In 2003, Bartholdi <a href="chapBib_mj.html#biBBartholdi03">[Bar03]</a> introduced the notion of an <em>L-presentation</em> for presentations of this type; that is, a group presentation of the form</p>
<p class="center">\[G=\left\langle S \left| Q\cup \bigcup_{\varphi\in\Phi^*} R^\varphi\right.\right\rangle,\]</p>
<p>where <span class="SimpleMath">\(\Phi^*\)</span> denotes the free monoid generated by a set of free group endomorphisms <span class="SimpleMath">\(\Phi\)</span>. 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 <span class="SimpleMath">\(n\)</span>-Engel groups).</p>
<p>The <strong class="pkg">lpres</strong>-package defines new <strong class="pkg">GAP</strong> 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 finitelyL-presented group <span class="SimpleMath">\(G\)</span> and a positive integer <span class="SimpleMath">\(c\)</span>. It computes a polycyclic presentation for the lower central series quotient <span class="SimpleMath">\(G/\gamma_{c+1}(G)\)</span>. Therefore, a nilpotent quotient algorithm can be used to determine the abelian invariants of the lower central series sections <span class="SimpleMath">\(\gamma_c(G)/\gamma_{c+1}(G)\)</span> and the largest nilpotent quotient of <span class="SimpleMath">\(G\)</span> if it exists.</p>
<p>Our nilpotent quotient algorithm generalizes Nickel's algorithm for finitely presented groups (see [Nic96]) which is implemented in the NQ-package; see [Nic03]. In difference to the NQ-package, the lpres-package is implemented in GAP only.
<p>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.
<p>A detailed description of our algorithm can be found in <a href="chapBib_mj.html#biBBEH08">[BEH08]</a> or in the diploma thesis <a href="chapBib_mj.html#biBH08">[Har08]</a>. Furthermore the <strong class="pkg">lpres</strong>-package includes the algorithms of <a href="chapBib_mj.html#biBMR2678952">[Har10]</a> for approximating the Schur multiplier of finitely L-presented groups.</p>
<p>The <strong class="pkg">lpres</strong>-package also includes the Reidemeister-Schreier algorithm from <a href="chapBib_mj.html#biBMR2876891">[Har12]</a>. L-presented groups were introduced as a tool to understand self-similar groups such as the Grigorchuk group. As such, <strongclass="pkg">lpres</strong> works in close contact with the package <strong class="pkg">fr</strong>. See <a href="chapBib_mj.html#biBMR3133711">[Har13]</a> for more on the relationships between L-presented groups and self-similar groups.</p>
<p>Finally, we note that we use the term "algorithm" somewhat loosely: many of the algorithms in this package are in fact semi-algorithms, guaranteed to give a correct answer but not guaranteed to terminate.</p>
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