Tangent groupoid and tangent cones in sub-Riemannian geometry

Let $X_1,\cdots ,X_m$ be vector fields satisfying Hörmander’s Lie bracket generating condition on a smooth manifold $M$. We generalise Connes’s tangent groupoid, by constructing a completion of the space $M\times M\times {\mathbb{R}}_{+}^{\times }$ 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 generalises a result of Bellaïche.