The Farrell-Jones Conjecture for the Hecke algebras of reductive p-adic groups

We formulate and sketch the proof of the K-theoretic Farrell-Jones Conjecture for the Hecke algebras of reductive p-adic groups. This is the first time that a version of the farrell-Jones Conjecture for topological groups is formulated. It implies that
the reductive projective class group of the Hecke algebra of a reductive p-adic group is the colimit of these for all compact open subgroups. This has been proved rationally by Bernstein and Dat using representation theory. The main applications of our result will concern the theory of smooth representations In particular we will prove a conjecture of Dat.

The proof is much more involved than the one for instance for discrete CAT(0)-groups. We will only give a very brief sketch of it focusing on the input from equivariant homotopy theory and discussing the new problems occurring in the setting of
totally disconnected groups. Most of the talk will be devoted an introduction to the Farrell-Jones Conjecture and the theory of
smooth representations of reductive p-adic groups, and discussion of  applications.

This is a joint project with Arthur Bartels.