The coarse assembly injectivity conjecture for bounded-geometry spaces

Let $X$ be a discrete metric space with bounded geometry. For each $d\geq 0$, let $P_d(X)$ be its Rips complex, and consider the assembly map

$$\mu:\lim_{d\to\infty}K_*(P_d(X))\to\lim_{d\to\infty}K_*(C^*(P_d(X)))\cong K_*(C^*(X)).$$

The coarse assembly injectivity conjecture. The map $\mu$ is injective. Injectivity of the coarse assembly map is the coarse Novikov property; the paper establishes this property for spaces with an FCE-by-FCE structure, while the general assertion remains open.