K- and L-theory of Group Rings

Abstract

Abstract. This article will explore the K-and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K-and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its consequences are surveyed. Mathematics Subject Classification (2000). Primary 18F25; Secondary 57XX.Keywords. K-and L-theory, group rings, Farrell-Jones Conjecture, topological rigidity. IntroductionThe algebraic K-and L-theory of group rings -K n (RG) and L n (RG) for a ring R and a group G -are highly significant, but are very hard to compute when G is infinite. The main ingredient for their analysis is the Farrell-Jones Conjecture. It identifies them with certain equivariant homology theories evaluated on the classifying space for the family of virtually cyclic subgroups of G. Roughly speaking, the Farrell-Jones Conjecture predicts that one can compute the values of these K-and L-groups for RG if one understands all of the values for RH, where H runs through the virtually cyclic subgroups of G.Why is the Farrell-Jones Conjecture so important? One reason is that it plays an important role in the classification and geometry of manifolds. A second reason is that it implies a variety of well-known conjectures, such as the ones due to Bass, Borel, Kaplansky and Novikov. (These conjectures are explained in Section 1.) There are many groups for which these conjectures were previously unknown but are now consequences of the proof that they satisfy the Farrell-Jones Conjecture. A third reason is that most of the explicit computations of K-and L-theory of group rings for infinite groups are based on the Farrell-Jones Conjecture, since it identifies them with equivariant homology groups which are more accessible via standard tools from algebraic topology and geometry (see Section 5).The rather complicated general formulation of the Farrell-Jones Conjecture is given in Section 3. The much easier, but already very interesting, special case of a torsionfree group is discussed in Section 2. In this situation the K-and L-groups are identified with certain homology theories applied to the classifying space BG. * The work was financially supported by the Leibniz-Preis of the author. The author wishes to thank several members and guests of the topology group in Münster for helpful comments.

Coarse flow spaces for relatively hyperbolic groups

“…We will review the precise formulation of the conjecture in § 4. More information about the Farrell–Jones conjecture and its applications can be found for example in [BLR08c, Lue10, LR05].…”

On Proofs of the Farrell–Jones Conjecture

“…The conjecture is related to a number of other conjectures in geometric topology and K-theory, most prominently the Borel Conjecture. Detailed discussions of applications and the formulation of this conjecture (and related conjectures) can be found in [10,32,33,34,35].…”

Algebraic K-theory, assembly maps, controlled algebra, and trace methods

“…There are other survey articles about the Farrell-Jones and related conjectures: [Bar16], [LR05], and [Mad94], which we already recommended, and also [Lüc10] and the voluminous book project [Lüc]. Our hope is that this contribution may serve as a more concise and accessible starting point, preparing the reader for these other more advanced surveys and for the original articles.…”