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.