The Higson–Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map

Abstract

We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson–Roe short exact sequence. When the action is co-compact, we show that the Roe $C^{\ast }$-ideal of locally compact operators is Morita equivalent to the reduced $C^{\ast }$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum–Connes map, via a Paschke–Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any $G$-equivariant Dirac-type family, a coarse index class which generalizes the Paterson index class and also the Moore–Schochet Connes’ index class for laminations.