Affine groups acting properly discontinuously. Mathematical developments arising from Hilbert's 18th problem.

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 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 $\mathbb{R}^n$ with compact quotient $\Gamma\backslash\mathbb{R}^n$ 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 in The structure of complete locally affine manifolds.

The Auslander Conjecture (1964). Every crystallographic subgroup $\Gamma$ of $\mathrm{GL}_n(\mathbb{R})$
is virtually solvable, i.e. contains a solvable subgroup of finite index.

Auslander's proof of this conjecture is unfortunately incorrect, but the conjecture is still an open and central problem:

Question (J. Milnor, 1976).  Does there exist a complete affinely flat manifold $M$ such that $\pi_1(M)$ contains a free group ?

We would like to explain the main ideas and methods, recent and old, behind our joint results with H. Abels and G. Margulis related to the above problems.