  
  [1X1 [33X[0;0YThe [5Xlpres[105X[101X[1X package[133X[101X
  
  [33X[0;0YThis  package  was  written  by  René  Hartung in 2009; maintenance has been
  overtaken by Laurent Bartholdi, who translated the manual to [5XGAPDoc[105X.[133X
  
  
  [1X1.1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YIn  1980,  Grigorchuk  [Gri80]  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
  [13XGrigorchuk group[113X 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 [Gri83])
  and  was  used  in  the  construction of a finitely presented amenable group
  which is not elementary amenable (see [Gri98]).[133X
  
  [33X[0;0YThe  Grigorchuk group is not finitely presentable (see [Gri99]). However, in
  1985,   Igor  Lysenok  (see  [Lys85])  determined  the  following  recursive
  presentation for the Grigorchuk group:[133X
  
  
  [24X[33X[0;6Y\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\inℕ)\rangle,[133X
  
  [124X
  
  [33X[0;0Ywhere  [22Xσ[122X  is  the  homomorphism  of  the  free group over [22X{a,b,c,d}[122X which is
  induced by [22Xa↦ c^a, b↦ d, c↦ b[122X, and [22Xd↦ c[122X. Hence, the infinitely many relators
  of this recursive presentation can be described in finite terms using powers
  of the endomorphism [22Xσ[122X.[133X
  
  [33X[0;0YIn  2003,  Bartholdi  [Bar03] introduced the notion of an [13XL-presentation[113X for
  presentations of this type; that is, a group presentation of the form[133X
  
  
  [24X[33X[0;6YG=\left\langle      S      \left|      Q\cup      \bigcup_{\varphi\in\Phi^*}
  R^\varphi\right.\right\rangle,[133X
  
  [124X
  
  [33X[0;0Ywhere  [22XΦ^*[122X  denotes  the  free  monoid  generated  by  a  set  of free group
  endomorphisms   [22XΦ[122X.  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 [22Xn[122X-Engel groups).[133X
  
  [33X[0;0YThe  [5Xlpres[105X-package defines new [5XGAP[105X 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  [22XG[122X  and  a  positive  integer  [22Xc[122X.  It  computes a
  polycyclic  presentation  for  the lower central series quotient [22XG/γ_c+1(G)[122X.
  Therefore,  a  nilpotent  quotient  algorithm  can  be used to determine the
  abelian  invariants of the lower central series sections [22Xγ_c(G)/γ_c+1(G)[122X and
  the largest nilpotent quotient of [22XG[122X if it exists.[133X
  
  [33X[0;0YOur nilpotent quotient algorithm generalizes Nickel's algorithm for finitely
  presented  groups  (see [Nic96]) which is implemented in the [5XNQ[105X-package; see
  [Nic03].  In  difference to the [5XNQ[105X-package, the [5Xlpres[105X-package is implemented
  in [5XGAP[105X only.[133X
  
  [33X[0;0YSince  finite L-presentations generalize finite presentations, our algorithm
  also  applies  to  finitely  presented  groups.  It  coincides with Nickel's
  algorithm in this special case.[133X
  
  [33X[0;0YA  detailed  description  of our algorithm can be found in [BEH08] or in the
  diploma   thesis   [Har08].   Furthermore  the  [5Xlpres[105X-package  includes  the
  algorithms  of  [Har10]  for  approximating the Schur multiplier of finitely
  L-presented groups.[133X
  
  [33X[0;0YThe  [5Xlpres[105X-package  also  includes  the Reidemeister-Schreier algorithm from
  [Har12].  L-presented  groups  were  introduced  as  a  tool  to  understand
  self-similar  groups  such  as the Grigorchuk group. As such, [5Xlpres[105X works in
  close contact with the package [5Xfr[105X. See [Har13] for more on the relationships
  between L-presented groups and self-similar groups.[133X
  
  [33X[0;0YFinally,  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.[133X
  
