Geometry of crystallographic groups. (Q2915833)

An \(n\)-dimensional \textit{crystallographic group} is a cocompact discrete subgroup of the group of isometries of Euclidean \(n\)-space. The algebraic structure of crystallographic groups is described by the classical Bieberbach theorems. Torsion free crystallographic groups are called \textit{Bieberbach groups}. Given a Bieberbach group \(\Gamma\) we can define an \(n\)-dimensional flat manifold \(M\) as quotient space \(\mathbb R^n/\Gamma\), and any compact flat manifold arises this way. The correspondence between geometric properties of flat manifolds and algebraic properties of the Bieberbach groups is one of the main themes of this book.NEWLINENEWLINE The contents of the book can be briefly described as follows. The first two chapters contain basic definitions and the proofs of the Bieberbach theorems. By the third Bieberbach theorem there are only finitely many isomorphism classes of crystallographic groups in any given dimension \(n\). Therefore one can look for their possible classification. Chapter 3 introduces three general classification methods: Zassenhaus' algorithm, Calabi's method and Auslander-Vasquez method. It also contains some classical classification results in dimensions \(2\) and \(3\). The next two chapters are related to the Calabi method, in particular, the author discusses here results related to flat manifolds with zero first Betti number and the outer automorphism groups of the crystallographic groups. Chapter 6 discusses the spin structure on a flat manifold and its application to the spectrum of the Dirac operator on the manifold. Chapter 7 gives necessary and sufficient conditions for the existence of a Kähler structure on an even dimensional flat manifold and presents some classification results of such manifolds in small dimensions. In Chapter 8 we find a theorem of D.~Long and A.~Reid showing that all flat manifolds are cusps of hyperbolic orbifolds. In Chapter 9 the author introduces a special class of crystallographic groups, called the generalized Hantzsche-Wendt groups, and studies its properties. The last chapter has a list of conjectures and open problems. The book ends with three appendixes treating (A) M.~Gromov's proof of the first Bieberbach theorem following the exposition by P.~Buser and H.~Karcher; (B) the proof of the Burnside transfer theorem; and (C) R.~Waldmüller's example of a flat manifold without symmetry.NEWLINENEWLINE The author emphasizes that most of the results discussed in this book were obtained after the publication of the influential monograph by \textit{L.~S.~Charlap} [Bieberbach groups and flat manifolds. New York: Springer-Verlag (1986; Zbl 0608.53001)].