The fundamental group of an affine manifold. Mathematical developments arising from Hilbert 18th problem.

Sun, 04/06/2017 - 14:00

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.