Baum–Connes conjecture for exact étale groupoids

Determine whether, for every exact second countable Hausdorff étale groupoid G, the Baum–Connes assembly map HG: Ktop(G) -> K*(C*(G)) is bijective, i.e., whether the Baum–Connes conjecture holds for exact étale groupoids.

Background

The paper reviews the Baum–Connes conjecture in the setting of étale groupoids, where the conjecture asserts that the Baum–Connes assembly map from topological K-theory to operator K-theory is an isomorphism. It is known to hold for groupoids with the Haagerup property (including amenable groupoids).

Despite these positive results, there are counterexamples to the conjecture in full generality. Within this landscape, the status of the conjecture for the important class of exact étale groupoids remains unresolved, and the authors explicitly note this open case.