Chern character for torsion-free ample groupoids

Abstract

For an ample groupoid with torsion-free stabilizers, we construct a Chern character map going from the domain of the Baum-Connes assembly map of G to the Crainic-Moerdijk homology groups of G with rational coefficients. Assuming the (rational) Baum-Connes conjecture, this implies that the operator K-groups of the groupoid C*-algebra are rationally isomorphic to the periodicized groupoid homology groups, confirming a variation of Matui's HK conjecture. Our construction hinges on the recent $\infty$-categorical viewpoint on bivariant K-theory. We also present applications to the homology of hyperbolic dynamical systems, the homology of topological full groups, the homotopy type of the algebraic K-theory spectrum of ample groupoids, and the Elliott invariant of classifiable C*-algebras.