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

Abstract

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 $\Gamma$, we introduce the equivariant coarse groupoid $G(X, \Gamma)$. We prove that the groupoid Baum--Connes conjecture for $G(X, \Gamma)$ with coefficients in $\ell^{\infty}(X,\mathcal{K})^\Gamma$ is equivalent to the equivariant coarse Baum--Connes conjecture for $(X, \Gamma)$ using a localization algebra description of equivariant $KK^\mathcal{G}$-theory for \'{e}tale 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 $\Gamma$-equivariant), then the equivariant coarse Novikov conjecture holds for $(X, \Gamma)$, i.e., the assembly map $\mu_{X,\Gamma}$ 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.