On the Farrell-jones Conjecture for Higher Algebraic K-theory

Here BΓ is the classifying space of the group Γ, and we denote by K−∞(R) the non-connective algebraic K-theory spectrum of the ring R. The homotopy groups of this spectrum are denoted Kn(R) and coincide with Quillen’s algebraic K-groups of R [Qui73] in positive dimensions and with the negative K-groups of Bass [Bas68] in negative dimensions. The homotopy groups of the spectrum X+∧K(R) are denoted Hn(X;K(R)). They yield a generalized homology theory and, in particular, standard computational tools such as the Atiyah-Hirzebruch spectral sequence apply to the left-hand side of the assembly map above. As a corollary of the main result of this paper we prove Conjecture 1.1 in the case where Γ is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. In fact our result is more general and applies to group rings RΓ, where R is a completely arbitrary coefficient ring. Note that if one replaces in Conjecture 1.1 the coefficient ring Z by an arbitrary coefficient ringR the corresponding conjecture would be false already in the simplest non-trivial case: if Γ = C is the infinite cyclic group the Bass-Heller-Swan formula [BHS64], [Gra76, p. 236] for Kn(RC) = Kn(R[t]) yields that Kn(RC) ∼= Kn−1(R)⊕Kn(R)⊕NKn(R)⊕NKn(R), where NKn(R) is defined as the cokernel of the split inclusion Kn(R) → Kn(R[t]) and does not vanish in general. But since S is a model for BC one obtains on the left-hand side of the assembly map only Hn(BC;K(R)) ∼= Kn(R)⊕Kn−1(R).