On homological coherence of discrete groups

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.