The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups

By Arthur Bartels, F. Thomas Farrell, W. Lück. 2011

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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 Kand 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. The Farrell-Jones Conjecture is not only important for calculations, but also gives structural insight, since the isomorphism occurring in its formulation has also geometric interpretations. This has the consequence that the Farrell-Jones Conjecture implies a variety of other well-known conjectures such as the ones due to Bass, Borel, Kaplansky, Novikov, and Serre which concern character theory for infinite groups, algebraic topology, the classification of manifolds, the ring structure of group rings, and group theory. We will discuss them in more detail in Subsection 2.2. The main result of this paper is to prove the Farrell-Jones Conjecture for a new prominent classes of groups, namely, cocompact lattices in almost connected Lie groups. We mention that the operator theoretic analog of the Farrell-Jones Conjecture, the Baum-Connes Conjecture, is known only for a few groups in this class. With the exception of the Novikov Conjecture, the conjectures listed above have not been known for this class so that our result presents also new contributions to them. Since we address a general version of the Farrell-Jones Conjectures, where one allows coefficients in additive categories, very powerful inheritance properties are valid which we will describe in Subsection 2.3. For instance, if this general version of the Farrell-Jones Conjecture holds for a group, it holds automatically for

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/33910267f6fa40ab87f663f363bbc5105a88d215

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Citations in the corpus

The Farrell–Jones conjecture for arbitrary lattices in virtually connected Lie groups (2014, background): “Let FJ be the class of groups satisfying the K- and L-theoretic Farrell–Jones conjecture with coefficients in additive categories and with finite wreath products.”
3-Manifold Groups (2012, background): “valent. On the other hand, homotopy equivalent lens spaces are not necessarily simple homotopy equivalent. We refer to [Mil66, Coh73, Rou11] and [Ki97, p. 119] for more details. Bartels–Farrell–Lu¨ck [BFL11] showed that the fundamental group of any 3-manifold Nsatisﬁes the Farrell–Jones Conjecture. ”