The Higson-Roe sequence for \'{e}tale groupoids. I. Dual algebras and compatibility with the BC map

TL;DR

This paper develops dual Roe algebras for proper étale groupoid actions, establishing a Higson-Roe sequence, and relates the boundary map to the Baum-Connes map, with applications to index theory for manifolds with groupoid symmetries.

Contribution

It introduces dual Roe algebras for étale groupoids, constructs the Higson-Roe sequence, and links the boundary map to the Baum-Connes map, extending index theory to groupoid actions.

Findings

Morita equivalence between Roe $C^*$-ideal and reduced $C^*$-algebra for cocompact actions

Identification of boundary map with Baum-Connes map via Paschke-Higson map

Definition of coarse index class for $G$-equivariant Dirac families on manifolds with bounded geometry

Abstract

We introduce the dual Roe algebras for proper \'{e}tale groupoid actions and deduce the expected Higson-Roe short exact sequence. When the action is cocompact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-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.