Measure contraction properties of Carnot groups

We prove that any corank 1 Carnot group of dimension k+ 1 equipped with a left-invariant measure satisfies the MCP (K, N) if and only if K≤ 0 and N≥ k+ 3. This generalizes the well known result by Juillet for the Heisenberg group Hk+1 to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number k+ 3 coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent [the least N such that the MCP (0 , N) is satisfied]. We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat.