The adiabatic groupoid and the Higson–Roe exact sequence

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