Arthur Bartels

The K -theoretic Farrell–Jones conjecture for hyperbolic groups

We prove the K-theoretic Farrell–Jones conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.

On the Isomorphism Conjecture in algebraic K -theory

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.

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

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

On the domain of the assembly map in algebraic K {theory

We compare the domain of the assembly map in algebraic K { theory with respect to the family of nite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later. AMS Classication 19D50; 19A31, 19B28

Squeezing and Higher Algebraic K-Theory

We prove that the Assembly map in algebraic Ktheory is split injective for groups of finite asymptotic dimension admitting a finite classifying space.

Inheritance of isomorphism conjectures under colimits

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.We present a

On the Farrell-Jones conjecture for higher algebraic -theory

C^*

All two dimensional links are null homotopic

-algebra which is naturally associated to the

Induction Theorems and Isomorphism Conjectures for K- and L-Theory

ax+b

Geodesic flow for CAT(0)–groups

-semigroup over