Algebraic K-theory of Geometric Groups

Abstract

In this paper we introduce a homotopy theoretic technique for proving that the $K$-theoretic assembly map is an equivalence. It is an extension of the methods used to prove split injectivity of the assembly and applies to any geometrically finite group. Our result is that there are two requirements which need to hold. The first is that the assembly map for the group regarded as a metric space is an equivalence. This is a non-equivariant condition and depends only on the coarse type of the word metric on the group. The second is that the group ring satisfies an algebraic coherence condition, which currently can be verified for all known groups for which the split injectivity statement for the assembly holds. The two conditions extend very broadly. In particular, both conditions hold for groups of finite asymptotic dimension. To state the main theorem precisely, given a regular Noetherian ring $A$ of finite global dimension and a group $\Gamma$ with finite $K(\Gamma,1)$ and finite asymptotic dimension, we prove that the $K$-theoretic assembly map is an equivalence. Therefore, in all dimensions the $K$-theory of $A[\Gamma]$ is the group homology of $\Gamma$ with coefficients in the $K$-theory spectrum of $A$. One of the many geometric consequences of this theorem is vanishing of the Whitehead group of $\Gamma$.