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1 The Almost Crystallographic Groups Package


  1. More about almost crystallographic groups

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 R3 or R4. That is one reason to define the dimension of an almost crystallographic group as its Hirsch length.

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.

1.1 More about almost crystallographic groups

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 3×3--matrices with real entries is of this type. Then L\rtimes Aut(L) acts affinely (on the left) on L via
l,l′ ∈ L,∀α ∈ Aut(L):  (l,α)l′=l  α(l′)·

Let C be a maximal compact subgroup of Aut(L). Then a subgroup G of L \rtimes C is said to be an almost crystallographic group if and only if the action of G on L, induced by the action of L\rtimes Aut(L), is properly discontinuous and the quotient space G \L is compact. One recovers the situation of the ordinary crystallographic groups by taking L=\Bbb Rn, for some n, and C=O(n), the orthogonal group.

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 \rtimes C. 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 G is its unique maximal normal nilpotent subgroup.

Theorem. The following are equivalent for a polycyclic-by-finite group G:

G is an almost crystallographic group.
Fitt(G) is torsion free and of finite index in G.
G contains a torsion free nilpotent normal subgroup N of finite index in G with CG(N) torsion free.
G has a nilpotent subgroup of finite index and there are no non-trivial finite normal subgroups in G.

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.

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aclib manual
January 2020