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

created by barilari on 13 Feb 2017
modified by rizzi1 on 06 Nov 2017

[BibTeX]

Accepted Paper

Inserted: 13 feb 2017
Last Updated: 6 nov 2017

Journal: Communications in Contemporary Mathematics
Year: 2017

ArXiv: 1702.04401 PDF

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.

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