The equivariant coarse Baum-Connes conjecture for actions by a-T-menable groups

The equivariant coarse Baum-Connes conjecture was firstly introduced by Roe [29] as a unified way to approach both the Baum-Connes conjecture and its coarse counterpart. In this paper, we prove that if an a-T-menable group $\Gamma$ acts properly and isometrically on a bounded geometry metric space $X$ with controlled distortion such that the quotient space $X/\Gamma$ is coarsely embeddable, then the equivariant coarse Baum-Connes conjecture holds for this action. This answers a question posed in [8] affirmatively.