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arXiv · 1801.06040

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

Moulay-Tahar Benameur·Indrava Roy
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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.

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Moulay-Tahar Benameur, Indrava Roy. 2018-01-18. The Higson-Roe sequence for \'{e}tale groupoids. I. Dual algebras and compatibility with the BC map. https://arxiv.org/abs/1801.06040

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