Weak coherence of groups and finite decomposition complexity

Abstract

The weak regular coherence is a coarse property of a finitely generated group $\Gamma$. 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 $K(\Gamma,1)$ 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 $K(\Gamma,1)$ and FDC. In particular, the Whitehead group $Wh (\Gamma)$ is trivial for such groups.