The adiabatic groupoid and the Higson–Roe exact sequence

Abstract

Let $\tilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric, and cocompact action of a discrete group $\Gamma$. In this paper, we prove that the analytic surgery exact sequence of Higson–Roe for $\tilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\tilde{X}{\times }_{\Gamma }\tilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $\varrho$-classes associated to a metric with a positive scalar curvature defined by Piazza and Schick (2014) correspond to the $\varrho$-classes defined by Zenobi (2019).