Wolfgang Lück: The Farrell-Jones Conjecture and its applications

Abstract: We give an introduction to the Farrell-Jones Conjecture which aims at the algebraic K- and L-theory of group rings. It is analogous to the Baum-Connes Conjecture about the topological K-theory of reduced group C*-algebras. We report on the substantial progress about the Farrell-Jones Conjecture which was made in the last years, it is meanwhile known for hyperbolic groups, CAT(0)-groups, S-arithmetic groups and lattices in almost connected Lie groups. We give a survey on its applications, for instance to the Novikov Conjecture, the Borel Conjecture and the classification of hyperbolic groups with a sphere of dimension greater or equal to five as boundary, Poincare duality groups, topological rigidity and so on. We will also give an outline of some work in progress concerning Hecke algebras and a stable version of the Cannon Conjecture.