Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures

Abstract: We prove that the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) can fail in sub-Riemannian geometry. This may happen beyond step two, if the distance function is not Lipschitz in charts, and it already occurs in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results, of independent interest, on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. As an application, we find sub-Riemannian structures with pre-medium-fat distribution that do not satisfy the MCP, answering a question raised in [L. Rifford, J. \'Ec. polytech. Math. 2023]. Finally, and quite unexpectedly, we show that ideal sub-Riemannian structures can fail the MCP, and this actually happens generically for rank greater than 3 and high dimension.