Geometry of crystallographic groups

Author(s): 
Andrzej Szczepa ́ski
Publisher: 
World Scientific
Year: 
2012
ISBN: 
978-981-4412-25-4
Short description: 

An $n$-dimensional crystallographic group $\Gamma$ is a discrete subgroup of the group $O(n)\ltimes{\mathbb R}$ of isometries of ${\mathbb R}^n$ having a compact fundamental domain. If $\Gamma$ is torsion free the quotient $M:={\mathbb R}^n/\Gamma$ is a manifold whose fundamental group is $\Gamma$, and since this group acts on ${\mathbb R}^n$ as a group of isometries, $M$ inherits a Riemannian structure making it into a flat manifold, i.e. a manifold with sectional curvature zero. Conversely, any compact flat manifold is obtained in this way, and many parts of this excellent book can be understood as a dictionary explaining the relationship between the geometric properties of $M$ and the algebraic properties of $\Gamma$.

MSC main category: 
20 Group theory and generalizations
MSC category: 
20H15
Other MSC categories: 
53C55
Reviewer: 
José Manuel Gamboa Mutuberria
Affiliation: 
Universidad Complutense de Madrid
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