The Baum–Connes Conjecture and Discrete Group Actions on Trees

We introduce a property (BC′) for discrete groups, which we prove to imply the Baum–Connes conjecture with coefficients and the K-amenability of the group. Then, we show that if Γ is a discrete group which acts on a tree X such that X/Γ is compact, and the stabilizers of the vertices and the stabilizers of the edges satisfy (BC′), then Γ itself satisfies (BC′) Finally, we indicate a couple of applications.