Title: Sharp measure contraction property for generalized H-type Carnot groups,
Abstract: We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the
$\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. 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.