The Integral K -theoretic Novikov Conjecture for Groups with Finite Asymptotic Dimension

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 V , V . The 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 ∪ . . .∪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 with a word metric. Examples from this apparently very large class are the Gromov hyperbolic groups [9], Coxeter groups [8], various generalized products of these, including the groups acting on trees with vertex stabilizers of finite asymptotic dimension [2], and, more generally, fundamental groups of developable complexes of finite dimensional groups [1]. We proved in [5] that cocompact lattices in connected Lie groups also have finite asymptotic dimension. Let K(A) be the nonconnective K-theory spectrum of the ring A. A discrete group is called geometrically finite if its classifying space has the homotopy type of a finite complex. Our main result is the following theorem. Main Theorem. Let Γ be a geometrically finite group with finite asymptotic dimension and letR be an arbitrary ring. Then the assemblymapα : h(Γ , K(R))→ K(R[Γ]) from the homology of the group Γ with coefficients in the K-theory spectrum K(R) to the K-theory of the group ring R[Γ] is a split injection. We shouldmention that the original Novikov conjecture on homotopy invariance of higher signatures has been verified for fundamental groups with finite asymptotic dimension by G. Yu [10]. Also, Gromov has constructed examples of geometrically finite groups with infinite asymptotic dimension, cf. [7], footnote to Problem 8 in section 9. The authors gratefully acknowledge support from the National Science Foundation.