THE FARRELL-JONES CONJECTURE FOR COCOMPACT LATTICES IN VIRTUALLY CONNECTED LIE GROUPS

摘要

1.1. Motivation and summary. The algebraic K-theory and L-theory of group rings has gained a lot of attention in the last decades, in particular since they play a prominent role in the classification of manifolds. Computations are very hard and here the Farrell-Jones Conjecture comes into play. It identifies the algebraic K-theory and L-theory of group rings with the evaluation of an equivariant homology theory on the classifying space for the family of virtually cyclic subgroups. This is the analogue of classical results in the representation theory of finite groups such as the induction theorem of Artin or Brauer, where the value of a functor for finite groups is computed in terms of its values on a smaller family, for instance of cyclic or hyperelementary subgroups; in the Farrell-Jones setting the reduction is to virtually cyclic groups. The point is that this equivariant homology theory is much more accessible than the algebraic K- and L-groups themselves. Actually, most of all computations for infinite groups in the literature use the Farrell-Jones Conjecture and concentrate on the equivariant homology side.