The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension

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 space X is called d-disjoint if dist(V, V ′) = inf{dist(x,x′)|x ∈ V, x ′ ∈ V ′}> d for all distinct subsets V, V ′. The asymptotic dimension ofX 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 =U0 ∪...∪Un. It is known that asymptotic dimension is a quasi-isometry invariant and so is an invariant of the finitely generated group, independent of the presentation. One says Γ has finite asymptotic dimension if it does as the metric space wit