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.