Sharp measure contraction property for generalized H-type Carnot groups

Abstract

We prove that H-type Carnot groups of rank [Formula: see text] and dimension [Formula: see text] satisfy the [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.