Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with proper, free, isometric and cocompact action of discrete group $\Gamma$. In this paper we prove that the analytic surgery exact sequence Higson-Roe for is isomorphic to associated adiabatic deformation Lie groupoid $\widetilde{X}\times_\Gamma\widetilde{X}$. We then generalize result context smoothly stratified manifolds. Finally,...