  
  [1X1 [33X[0;0YThe Almost Crystallographic Groups Package[133X[101X
  
  [33X[0;0YA  group  is  called  [13Xalmost  crystallographic[113X if it is a finitely generated
  nilpotent-by-finite  group  without  non-trivial finite normal subgroups. An
  important  special  case  of  almost  crystallographic groups are the [13Xalmost
  Bieberbach groups[113X: these are almost crystallographic and torsion free.[133X
  
  [33X[0;0YBy  its  definition,  an  almost  crystallographic  group  [22XG[122X  has a finitely
  generated  nilpotent  normal  subgroup  [22XN[122X  of  finite  index.  Clearly, [22XN[122X is
  polycyclic  and  thus has a polycyclic series. The number of infinite cyclic
  factors in such a series for [22XN[122X is an invariant of [22XG[122X: the [13XHirsch length[113X of [22XG[122X.[133X
  
  [33X[0;0YFor each almost crystallographic group of Hirsch length 3 and 4 there exists
  a   representation  as  a  rational  matrix  group  in  dimension  4  or  5,
  respectively.   These   representations   can   be   considered   as  affine
  representations  of  dimension 3 or 4. Via these representations, the almost
  crystallographic  groups  act (properly discontinuously) on [22Xℝ^3[122X or [22Xℝ^4[122X. That
  is one reason to define the [13Xdimension[113X of an almost crystallographic group as
  its Hirsch length.[133X
  
  [33X[0;0YThe  3-dimensional  and  a part of the 4-dimensional almost crystallographic
  groups  have  been  classified by K. Dekimpe in [Dek96]. This classification
  includes  all almost Bieberbach groups in dimension 3 and 4. It is the first
  central  aim  of  this  package  to  give access to the resulting library of
  groups. The groups in this electronic catalog are available in two different
  representations:  as  rational matrix groups and as polycyclically presented
  groups.  While  the first representation is the more natural one, the latter
  description  facilitates  effective  computations with the considered groups
  using the methods of the [5XPolycyclic[105X package.[133X
  
  [33X[0;0YThe  second  aim of this package is to introduce a variety of algorithms for
  computations  with  polycyclically presented almost crystallographic groups.
  These  algorithms supplement the methods available in the [5XPolycyclic[105X package
  and  give  access  to  some  methods  which are interesting specifically for
  almost crystallographic groups. In particular, we present methods to compute
  Betti  numbers and to construct or check the existence of certain extensions
  of  almost  crystallographic  groups.  We  note that these methods have been
  applied  in  [DE03] and [DE02] for computations with almost crystallographic
  groups.[133X
  
  [33X[0;0YFinally,  we  remark  that  almost  crystallographic  groups  can be seen as
  natural   generalizations   of   crystallographic   groups.   A  library  of
  crystallographic  groups  and  algorithms  to  compute with crystallographic
  groups are available in the [5XGAP[105X packages [5XCryst[105X, [5XCaratInterface[105X and [5XCrystCat[105X.[133X
  
  
  [1X1.1 [33X[0;0YMore about almost crystallographic groups[133X[101X
  
  [33X[0;0YAlmost crystallographic groups were first discussed in the theory of actions
  on  Lie  groups. We recall the original definition here briefly and we refer
  to [Aus60], [Dek96] and [Lee88] for more details.[133X
  
  [33X[0;0YLet  [22XL[122X be a connected and simply connected nilpotent Lie group. For example,
  the  3-dimensional  Heisenberg  group, consisting of all upper unitriangular
  [22X3×3[122X-matrices with real entries is of this type. Then [22XL⋊ Aut(L)[122X acts affinely
  (on the left) on [22XL[122X via[133X
  
  
  [24X[33X[0;6Y\forall   l,l'\in   L,\forall   \alpha   \in  Aut(L):  \;  ^{(l,\alpha)}l'=l
  \alpha(l').[133X
  
  [124X
  
  [33X[0;0YLet [22XC[122X be a maximal compact subgroup of [22XAut(L)[122X. Then a subgroup [22XG[122X of [22XL ⋊ C[122X is
  said  to  be an almost crystallographic group if and only if the action of [22XG[122X
  on  [22XL[122X, induced by the action of [22XL⋊ Aut(L)[122X, is properly discontinuous and the
  quotient  space [22XG ∖ L[122X is compact. One recovers the situation of the ordinary
  crystallographic  groups  by  taking  [22XL=ℝ^n[122X,  for  some  [22Xn[122X,  and [22XC=O(n)[122X, the
  orthogonal group.[133X
  
  [33X[0;0YMore  generally, we say that an abstract group is an almost crystallographic
  group if it can be realized as a genuine almost crystallographic subgroup of
  some   [22XL   ⋊   C[122X.  In  the  following  theorem  we  outline  some  algebraic
  characterizations  of  almost  crystallographic groups; see Theorem 3.1.3 of
  [Dek96].  Recall that the [13XFitting subgroup Fitt[22X(G)[122X[113X of a polycyclic-by-finite
  group [22XG[122X is its unique maximal normal nilpotent subgroup.[133X
  
  [33X[0;0Y[13XTheorem.[113X The following are equivalent for a polycyclic-by-finite group [22XG[122X:[133X
  
  [31X1[131X   [33X[0;6Y[22XG[122X is an almost crystallographic group.[133X
  
  [31X2[131X   [33X[0;6YFitt[22X(G)[122X is torsion free and of finite index in [22XG[122X.[133X
  
  [31X3[131X   [33X[0;6Y[22XG[122X  contains a torsion free nilpotent normal subgroup [22XN[122X of finite index
        in [22XG[122X with [22XC_G(N)[122X torsion free.[133X
  
  [31X4[131X   [33X[0;6Y[22XG[122X  has  a  nilpotent  subgroup  of  finite  index  and  there  are  no
        non-trivial finite normal subgroups in [22XG[122X.[133X
  
  [33X[0;0YIn  particular, if [22XG[122X is almost crystallographic, then [22XG / Fitt(G)[122X is finite.
  This factor is called the [13Xholonomy group[113X of [22XG[122X.[133X
  
  [33X[0;0YThe  dimension  of  an almost crystallographic group equals the dimension of
  the  Lie  group  [22XL[122X  above which coincides also with the Hirsch length of the
  polycyclic-by-finite  group.  This  library  therefore  contains families of
  virtually nilpotent groups of Hirsch length 3 and 4.[133X
  
