The coarse Baum--Connes conjecture and groupoids II

Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum–Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M) is an isomorphism, is hereditary by taking closed subspaces.

💡 Research Summary

The paper investigates the coarse Baum–Connes conjecture in the setting of groupoids, extending the authors’ previous work to a broader class of metric spaces. The authors start with a proper metric space (M) that satisfies bounded geometry, a condition guaranteeing uniform control over the size of metric balls and the existence of a uniformly finite covering by uniformly bounded subsets. For such a space they construct an étale groupoid (G(M)) as follows: for each radius (R>0) consider the relation \