Prof. Dr. Holger Reich, I. Melnikov
16.11.2010, 16:00 Uhr – 19:00 Uhr
Rigidity and dynamics
Prof. Dr. Holger Reich
16.11.2010, 16:00 Uhr – 17:00 Uhr
The Borel conjecture says that closed aspherical manifolds are
topologically rigid, i.e. if two such manifolds are homotopy
equivalent, then they are already homeomorphic. The Kaplansky conjecture says that there are no nontrivial idempotents in the group ring of a torsionfree group. The Bass conjecture is concerned with the prolongation of character theory from finite to infinite groups.
The talk will try to indicate how these and other conjectures are related to algebraic K-theory and how they are subsumed in the so called Farrell-Jones conjecture, which can be formulated for an arbitrary discrete group. The Farrell-Jones conjecture is known for many groups but it is completely open in general.
All known proofs of the Farrell-Jones conjecture assume that the group is acting by isometries on a suitable geometry, in the simplest case a Riemannian manifold. Dynamical properties of the geoemtry are used in the proofs.
We will give an overview of known results and indicate future directions.