KTH/SU Mathematics Colloquium
April 22, 2009
Herbert Abels, University of Bielefeld
Affine crystallographic and properly discontinuous groups
There is a long standing conjecture of Auslander (1964) which states that
every affine crystallographic group has a solvable subgroup of finite index.
Milnor (1977) asked if, more generally, every properly discontinuous affine
group has a solvable subgroup of finite index. This more general conjecture
was disproved by Margulis (1983). The Auslander conjecture has been proved
in many cases, but is wide open in dimension at least 7.
I will explain the geometric and algebraic relevance of the notions and conjectures and
will give an idea of the methods involved in the constructions and proofs in our joint work
with Margulis and Soifer. We use dynamics of linear and affine maps. The connection
with the Tits alternative and proximal maps will be pointed out.