(Locally) shortest arcs of special sub-Riemannian metric on the Lie   group $SO_0(2,1)$
  The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group $SO_0(2,1)$ under the condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO_0(2,1)$. 
Valera Berestovskii
math.DG