Tangent groupoid and tangent cones in sub-Riemannian geometry

Let X1,…,Xm be vector fields satisfying Hörmander’s Lie bracket generating condition on a smooth manifold M. We generalize Connes’s tangent groupoid by constructing a completion of the space M×M×R+× using the sub-Riemannian metric. We use our space to calculate all the tangent cones of the sub-Riemannian metric in the sense of the Gromov–Hausdorff distance. This generalizes a result of Bellaïche.