A group is called **almost crystallographic** 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 **almost
Bieberbach groups**: these are almost crystallographic and torsion free.

By its definition, an almost crystallographic group `G` has a finitely
generated nilpotent normal subgroup `N` of finite index. Clearly, `N` is
polycyclic and thus has a polycyclic series. The number of infinite cyclic
factors in such a series for `N` is an invariant of `G`: the **Hirsch length**
of `G`.

For 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 `R ^{3}` or

The 3-dimensional and a part of the 4-dimensional almost crystallographic groups have been classified by K. Dekimpe in KD. 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 Polycyclic package.

The 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 Polycyclic 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 DE1 and DE2 for computations with almost crystallographic groups.

Finally, 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 GAP packages `cryst`

, `carat`

and `crystcat`

.

Almost crystallographic groups were first discussed in the theory of actions on Lie groups. We recall the original definition here briefly and we refer to AUS, KD and LEE for more details.

Let `L` be a connected and simply connected nilpotent Lie group. For
example, the 3-dimensional Heisenberg group, consisting of all upper
unitriangular `3times3`--matrices with real entries is of this type.
Then `LrtimesAut(L)` acts affinely (on the left) on `L` via

` foralll,l'inL,forallalphainAut(L):;
^{(l,alpha)}l'=l , alpha(l'). `

Let `C` be a maximal compact subgroup of `Aut(L)`. Then a subgroup `G`
of `L rtimesC` is said to be an almost crystallographic group if and only
if the action of `G` on `L`, induced by the action of `LrtimesAut(L)`,
is properly discontinuous and the quotient space `G backslashL` is compact.
One recovers the situation of the ordinary crystallographic groups by taking
`L=BbbR ^{n}`, for some

More 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 `L rtimesC`. In the following theorem we outline some algebraic
characterizations of almost crystallographic groups; see Theorem 3.1.3 of
KD. Recall that the **Fitting subgroup Fitt (G)** of a
polycyclic-by-finite group

proclaimTheorem.
The following are equivalent for a polycyclic-by-finite group `G`:
parindent30pt

parindent0pt

In particular, if `G` is almost crystallographic, then `G / Fitt(G)`
is finite. This factor is called the **holonomy group** of `G`.

The dimension of an almost crystallographic group equals the dimension
of the Lie group `L` 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.

aclib manual

Mai 2012