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Almost-Bieberbach groups: affine and polynomial structures. (English) Zbl 0865.20001

Lecture Notes in Mathematics. 1639. Berlin: Springer. x, 259 p. (1996).
The conceptual foundation of this monograph is given by the \(n\)-dimensional crystallographic space groups, object of three famous theorems by Bieberbach in his work on the groups of rigid motions. The first theorem states that the translations in a space group \(\Pi\) form a lattice group \(\Gamma\) and that the factor group \(\Pi/\Gamma\) (isomorphic to the point group) is finite. According to its second theorem, two isomorphic space groups are affine conjugated and the third theorem implies that for a given dimension, and up to affine conjugation, there are only finitely many space groups.
The crystallographic space groups, beside in crystallography, play an important role in mathematics. The aspect considered in this book concerns the so-called Bieberbach groups, which are the torsion free space groups. These groups occur as fundamental groups of flat Riemannian manifolds which, therefore, can be characterized algebraically and fully classified. The point group then appears as the holonomy group.
The aim of the book is to discuss a generalization of space groups (called almost-crystallographic, or AC-groups), by replacing the abelian group of translations by a nilpotent group. The torsion-free case gives the almost-Bieberbach groups (the AB-groups). In the same way, the flat Riemannian manifolds are generalized to nilmanifolds (with an AC-group as fundamental group) and to infra-nilmanifolds in the AB-group case. The problem of an algebraic characterization and of a classification of these new manifolds amounts to the generalization of the three Bieberbach theorems. The first two are fairly straightforwards, the third one requires more concepts and in the book two alternative proofs are given: one by reducing the classification of AC-groups having a fixed maximal nilpotent subgroup to the known classification of space groups. The second one is based on a group extension with kernel the nilpotent group \(N\). A particularly important case is when the AC-group is a polycyclic-to-finite group with \(N\) the Fitting subgroup, i.e. the unique maximal normal nilpotent subgroup.
The definitions required for an algebraic characterization of the AB-groups form the first part of the monograph. A second part deals with group representations of a given canonical type of polycyclic-by-finite groups: in particular an affine one, involving the standard matrix representation of the affine group, and a polynomial one involving polynomial diffeomorphisms of \(R^K\) with polynomials in \(K\) variables. Finally, the third part is devoted to the classification, construction and characterization of 3-dimensional AC-groups and of 4-dimensional AB-groups with a 2-step nilpotent subgroup (maximal, normal, Fitting). The construction is based on several computational algorithms (implemented in Mathematica) allowing, in particular, the computation of second cohomology groups \(H^2(Q,Z)\) with \(Q\) a crystallographic group.
As the author stresses in the Preface, the richness of the mathematical properties of crystallographic groups should allow further generalizations. In my view the interest of this monograph is not only a mathematical one but also for new developments in crystallography. As it is well known, higher-dimensional space groups are used for the symmetry characterization of 3-dimensional aperiodic crystal structures (the incommensurate crystals and the quasicrystals) and non-abelian translation groups have already been considered by Kramer on the basis of free groups. I see this book as an invitation to consider AC-groups as an alternative possibility within a more comprehensive mathematical crystallography.

MSC:

20-02 Research exposition (monographs, survey articles) pertaining to group theory
20H15 Other geometric groups, including crystallographic groups
82D25 Statistical mechanics of crystals
30F10 Compact Riemann surfaces and uniformization
57S30 Discontinuous groups of transformations
20F18 Nilpotent groups
22E25 Nilpotent and solvable Lie groups
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