We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The new notion is defined in terms of the coarse geometry of groups and should be as useful for computing their K-theory. We prove that a group Γ of finite asymptotic dimension is weakly coherent. In particular, there is a large collection of R[Γ ]-modules of finite homological dimension when R is a finite-dimensional regular ring. This class contains word-hyperbolic groups, Coxeter groups and, as we show, the cocompact discrete subgroups of connected Lie groups.

Abstract. The integral assembly map in algebraic K-theory is split injective for any geometrically finite discrete group with finite asymptotic dimension.The goal of this paper is to apply the techniques developed by the first author in [3] to verify the integral Novikov conjecture for groups with finite asymptotic dimension as defined by M. Gromov [9].Recall that a finitely generated group Γ can be viewed as a metric space with the word metric associated to a given presentation. Definition (Gromov). A family of subsets in a general metric spaceThe asymptotic dimension of X is defined as the smallest number n such that for any d > 0 there is a uniformly bounded cover U of X by n + 1 d-disjoint families of subsets U = U 0 ∪ . . . ∪ U n .

We construct a new compactification of a noncompact rank one globally symmetric space. The result is a nonmetrizable space which also compactifies the Borel-Serre enlargement X of X, contractible only in the appropriateČech sense, and with the action of any arithmetic subgroup of the isometry group of X on X not being small at infinity. Nevertheless, we show that such a compactification can be used in the approach to Novikov conjectures developed recently by G. Carlsson and E. K. Pedersen. In particular, we study the nontrivial instance of the phenomenon of bounded saturation in the boundary of X and deduce that integral assembly maps split in the case of a torsion-free arithmetic subgroup of a semi-simple algebraic Q-group of real rank one or, in fact, the fundamental group of any pinched hyperbolic manifold. Using a similar construction we also split assembly maps for neat subgroups of Hilbert modular groups.

The weak regular coherence is a coarse property of a finitely generated group Γ. It was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen's regular coherence as part of computation of the integral K-theoretic assembly map.A new class of metric spaces (sFDC) was introduced recently by A. Dranishnikov and M. Zarichnyi. This class includes most notably the spaces with finite decomposition complexity (FDC) studied by E. Guentner, D. Ramras, R. Tessera, and G. Yu. The main theorem of this paper shows that a group that has finite K(Γ, 1) and sFDC is weakly regular coherent.As a consequence, the integral K-theoretic assembly maps are isomorphisms in all dimensions for any group that has finite K(Γ, 1) and FDC. In particular, the Whitehead group Wh(Γ) is trivial for such groups.

Let Γ be a geometrically finite group of finite asymptotic dimension and let R be a regular Noetherian ring of finite homological dimension. The main result of the paper is a proof of the Isomorphism Conjecture that the integral K-theoretic assembly map for the group ring R[Γ] is an isomorphism. We also include partial results for larger classes of geometric groups.