# 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 \text{Aff}$(\mathbb{R}^n)$, of crystallographic groups $\G $ 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 $\G$ 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 \bigskip \\

\pro {\it The Auslander Conjecture.} Every crystallographic subgroup $\Gamma$ of \text{Aff}$(\mathbb{R}^n)$

is virtually solvable, i.e. contains a solvable subgroup of finite

index. \endpro\\

In 1977 J. Milnor stated the following question:\\

\pro {\it Question.} Does there exist a complete affinely flat manifold $M$ such that $\pi_1(M) $ contains a free group ? \endpro \\

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

- Last modified: 28/05/2017