Fundamental groups of affine manifolds

The study of the fundamental group of an affine manifold has a long history that goes
back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact
Euclidian affine manifold has a subgroup of a finite index such that every element of this
subgroup is translation. The motivation was the study of the symmetry groups of crys-
talline structures which are of fundamental importance in the science of crystallography.
A natural way to generalize the classical problem is to broaden the class of allowed mo-
tions and consider groups of affine transformations. In 1964, L. Auslander in his paper
”The structure of complete locally affine manifolds” stated the following conjecture, now
known as the Auslander conjecture: The fundamental group of a compact complete locally
flat affine manifold is virtually solvable.
In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”,
J.Milnor asked if a free group can be the fundamental group of complete affine flat mani-
fold.
The purpose of the talk is to recall the old and to talk about our new results in light of
these questions. Our approach is based on the study of the dynamic of an affine action.

The talk is aimed at a wide audience and all notions will be explained.