The fundamental group of an affine manifold. Mathematical developments arising from Hilbert 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(R^n) of crystallographic groups G acting  isometrically on 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 G acting isometrically on  the n-dimensional Euclidean space R^n contains a subgroup  of finite index consisting of translations. In particular, such a group G is virtually abelian, i.e. G contains an abelian subgroup of finite index. In 1964 Auslander proposed  the following conjecture 

The Auslander Conjecture: Every crystallographic subgroup G of Aff(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 the fundamental group of M contains a free group?

We will explain ideas and methods, recent and old  results related to the above problems.