We prove the Borel Conjecture for a class of groups containing wordhyperbolic groups and groups acting properly, isometrically and cocompactly on a finite-dimensional CAT(0)-space.

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic n-space H n with finite volume orbit space. We then apply this result to show that, for any Bianchi group Γ, W h(Γ), ˜ K 0 (ZΓ), and K i (ZΓ) vanish for i ≤ −1.

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture for K .RG/ for every word-hyperbolic group G and every coefficient ring R.

Abstract. Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-dimensional manifold. We prove the K-and L-theoretic Farrell-Jones Conjecture for G.

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

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.