The Sub-Riemannian cut locus of $H$-type groups
  In the present paper we give a proof of the fact that the sub-Riemannian cut locus of a wide class of nilpotent groups of step two, called $H$-type groups, starting from the origin corresponds to the center of the group. We obtain this result by completely describing the sub-Riemannian geodesics in the group, and using these to obtain three disjoint sets of points in the group determined by the number of geodesics joining them to the origin. 
Christian Autenried, Mauricio Godoy Molina
math.DG