The uniform coarse Baum–Connes isomorphism conjecture for Rips complexes

Let $Y$ be a proper, uniformly discrete metric space with coarsely bounded geometry. For each $d\geq 0$, let $P_d(Y)$ be the Rips complex of $Y$, and let

$$\mu_u \colon \lim_{d \to \infty} K_\ast^u(P_d(Y)) \to K_\ast(C_u^\ast(Y))$$

be the uniform coarse assembly map.

Uniform coarse Baum–Connes conjecture. The map $\mu_u$ is an isomorphism. This is the discrete Rips-complex formulation of the uniform coarse Baum–Connes conjecture, connecting uniform $K$-homology of the large-scale approximations $P_d(Y)$ with the uniform Roe algebra of $Y$.