[aclib] 2 Algorithms for almost crystallographic groups

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# 2 Algorithms for almost crystallographic groups

### Sections

1. Properties of almost crystallographic groups
2. Betti numbers
3. Determination of certain extensions

This chapter presents a variety of algorithms for almost crystallographic groups. In most cases, they assume a polycyclically presented group as input; in particular, the input groups must be polycyclic in this case. The methods described here supplement the methods of the Polycyclic package for polycyclically presented groups. Many of the functions in this chapter are based on methods of the Polycyclic package and thus this package must be installed to use the functions introduced here. We refer to the Polycyclic package for further information on polycyclic presentations.

## 2.1 Properties of almost crystallographic groups

- `IsAlmostCrystallographic(` G`) P`

  This function checks if a polycyclically presented group G is almost crystallographic; that is, it checks if G is nilpotent-by-finite and has no non-trivial finite normal subgroup.

- `IsAlmostBieberbachGroup(` G`) P`

  This function checks if a polycyclically presented group G is almost Bieberbach; that is, it checks if G is nilpotent-by-finite and torsion free.

  ## 2.2 Betti numbers

  Let G be a polycyclically presented and torsion free group of Hirsch length n. Then we can compute the Betti numbers βi(G) for i ∈ {0, 1, 2, n−2, n−1, n}. If n ≤ 6, then we can compute all Betti numbers βi(G) for 0 ≤ i ≤ 6 of G. We introduce the following functions for this purpose and we refer to BRO for the details on the orientation module and the Betti numbers.

- `OrientationModule(` G`) F`

  This function determines the orientation module of the polycyclically presented group G; that is, it returns a list of matrices m1, …, mn ≤ GL( 1, Z) which are the images of the 'Igs(G)' in their action on the orientation module.

- `BettiNumber(` G`,` m`) F`

  This function returns the mth Betti number of the polycyclically presented torsion free group G if m ∈ {0, 1, 2, n−2, n−1, n}, where n is the Hirsch length of G.

- `BettiNumbers(` G`) A`

  This function returns the Betti number of the polycyclically presented torsion free group G if the Hirsch length of G is smaller than 7.

  ## 2.3 Determination of certain extensions

  Let G be a polycyclically presented almost crystallographic group. We want to check the existence of certain extensions of G.

  First, it is well-known that the equivalence classes of extensions of G correspond to the second cohomology group of G. This cohomology group can be computed using the methods of the Polycyclic package for any explicitly given module of G. Further, we can construct a polycyclic presentation for each cocycle of the second cohomology group. We give an example for such a computation below.

  However, we may be interested in certain extensions only; for example, the torsion free extensions are often of particular interest. If the second cohomology group is finite, then we can compute a polycyclic presentation for each element of this group and check the resulting group for torsion freeness. But if the second cohomology group is infinite, then this approach is not available. Hence we introduce the following special method to cover this and related applications.

- `HasExtensionOfType(` G`,` torsionfree`,` minimalcentre`) F`

  Suppose that G is a polycyclically presented almost crystallographic group with Fitting subgroup N. This function checks if there is a G-module M ≅ Z which is centralized by N such that there exists a torsion free extension of M by G (if the flag torsionfree is true) or an extension E with Z(Fitt(E)) = M (if the flag minimalcentre is true) or an extension which satisfies both conditions (if both flags are true).

  We note that the existence of such extensions is of interest in the determination of extensions which are almost Bieberbach groups. We refer to DE1 for a more detailed account of this application and for further results of a similar nature.

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