Let $widetilde{X}$ 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 $widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $widetilde{X}times_Gammawidetilde{X}$. 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 positive scalar curvature defined by Piazza and Schick corresponds to the $varrho$-classes defined by the author of this paper.
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