Reviewer:

José Manuel Gamboa Mutuberria

Affiliation:

Universidad Complutense de Madrid

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 $\Gamma$.

## Publisher:

Published:

2012

ISBN:

978-981-4412-25-4