Аннотация:
The study of affine crystallographic groups has a long history which goes back to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in $\mathrm{Aff}(\mathbb{R}^n)$, of crystallographic groups $\Gamma$ acting isometrically on $\mathbb{R}^n$. In a series of papers Bieberbach showed that this was so. The key result is the following famous theorem of Bieberbach. A crystallographic group $\Gamma$ acting isometrically on the $n$-dimensional Euclidean space $\mathbb R^n$ contains a subgroup of finite index consisting of translations. In particular, such a group $\Gamma$ is virtually abelian, i.e. $\Gamma$ contains an abelian subgroup of finite index. In 1964 Auslander proposed the following conjecture
The Auslander Conjecture. Every crystallographic subgroup $\Gamma$ of $\mathrm{Aff}(\mathbb{R}^n)$ is virtually solvable, i.e. contains a solvable subgroup of finite index.
In 1977 J. Milnor stated the following question:
Question. Does there exist a complete affinely flat manifold $M$ such that $\pi_1(M) $ contains a free group ?
We will explain ideas and methods, recent and old results related to the above problems.