A groupoid approach to the equivariant coarse Baum--Connes conjecture

In this paper, we develop a groupoid approach to the equivariant coarse Baum–Connes conjecture. For a bounded geometry metric space $X$ equipped with a proper, free, and isometric action of a countable discrete group $\mathrm{\Gamma }$, we introduce the equivariant coarse groupoid $G(X,\mathrm{\Gamma })$. We prove that the groupoid Baum–Connes conjecture for $G(X,\mathrm{\Gamma })$ with coefficients in ${\mathrm{\ell }}^{\mathrm{\infty }}(X,\mathcal{K}{)}^{\mathrm{\Gamma }}$ is equivalent to the equivariant coarse Baum–Connes conjecture for $(X,\mathrm{\Gamma })$ using a localization algebra description of equivariant $K{K}^{\mathcal{G}}$-theory for étale groupoids. As applications of this framework, we prove that if the space $X$ admits a coarse embedding into Hilbert space (which is not required to be $\mathrm{\Gamma }$-equivariant), then the equivariant coarse Novikov conjecture holds for $(X,\mathrm{\Gamma })$, i.e., the assembly map ${\mu }_{X,\mathrm{\Gamma }}$ is an injection. We also obtain a new proof of the equivariant coarse Baum–Connes conjecture if $X$ admits an equivariant coarse embedding into Hilbert space.