# Sharp measure contraction property for generalized H-type Carnot groups

## TL;DR

This paper establishes sharp measure contraction properties for generalized H-type Carnot groups, linking curvature bounds to the geodesic dimension and analyzing optimal synthesis and Wasserstein geodesics.

## Contribution

It extends MCP results to a broad class of generalized H-type Carnot groups, explicitly computing optimal synthesis and connecting curvature exponents to geodesic dimension.

## Key findings

- H-type Carnot groups satisfy MCP(K,N) iff K≤0 and N≥k+3(n−k)
- The class of generalized H-type groups includes all step 2 groups with abnormal minimizers
- Wasserstein geodesics are absolutely continuous on these 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.