Curvature exponent and geodesic dimension on Sard-regular Carnot groups

In this paper we characterize the geodesic dimension $N_{GEO}$ and give a new lower bound to the curvature exponent $N_{CE}$ on Sard-regular Carnot groups. As an application, we give an example of step-two Carnot group on which $N_{CE}>N_{GEO}$: this answers a question posed by Rizzi in arXiv:1510.05960v4 [math.MG].