1.1 The Isomorphism Conjecture in algebraic K-theory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6

Abstract. We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version of the Farrell-Jones Conjecture. 1.

Abstract. We prove the Farrell-Jones Isomorphism Conjecture for groups acting on complete hyperbolic manifolds with finite volume orbit space. We then apply this result to show that for any Bianchi group Γ, W h(Γ), ˜ K0(ZΓ), and Ki(ZΓ) vanish for i ≤ −1. 1.

Abstract. We redefine the Baum-Connes assembly map using simplicial approximation

Abstract. In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K0(kΓ) is proved to be isomorphic to the colimit of K0(kH) over the finite subgroups H of Γ, when Γ is a virtually polycyclic group and k is a field of characteristic zero. 1.

Abstract. 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 also added a brief epilogue, essentially “What there wasn’t time for. ” Although the focus of the conference was on noncommutative geometry, the topic discussed was conventional commutative motivations for the circle of ideas related to the Novikov and Baum-Connes conjectures. While the article is mainly expository, we present here a few new results (due to the two of us). It is interesting to note that while the period from 80’s through the mid-90’s has shown a remarkable convergence between index theory and surgery theory (or more generally, the classification of manifolds) largely motivated by the Novikov conjecture, most recently, a number of divergences has arisen. Possibly, these subjects are now diverging, but it also seems plausible that we are only now close to discovering truly deep phenomena and that the difference between these subjects is just one of these. Our belief is that, even after decades of mining this vein, the gold is not yet all gone. As the reader might guess from the title, the focus of these notes is not quite on the Novikov conjecture itself, but rather on a collection of problems that are suggested by heuristics, analogies and careful consideration of consequences. Many of the related conjectures are false, or, as far as we know, not directly mathematically related to the original conjecture; this is a good thing: we learn about the subtleties of the original problem, the boundaries of the associated phenomenon, and get to learn about other realms of mathematics.

For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O + (n, 1) = Isom(H n). We use the information we obtain to explicitly compute the lower algebraic K-theory of the Coxeter group Γ3 (a non-uniform lattice in O + (3, 1)). Part of this computation involves calculating certain Waldhausen Nil-groups for Z[D2], Z[D3].

Abstract. It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model for EΓ. In particular, it applies to discrete subgroups of virtually connected Lie groups.

Abstract. Let Γ be a cocompact Fuchsian group. We calculate the lower algebraic K-theory of the integral group ring ZΓ and find an explicit formula for Ki(ZΓ), i ≤ 1, in terms of the lower K-groups of finite cyclic groups. 1.