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 mathbbHk+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.