On symmetries of a sub--Riemannian structure with growth vector $(4,7)$

Abstract

We study the action of symmetries on geodesics of a control problem on a Carnot group with growth vector (4, 7). We show that there is a subgroup of symmetries isomorphic to SO(3) and a set of points in the Carnot group with a nontrivial stabilizer of this action. We prove that each geodesic either lies in this set or do not intersect this set. In the former case the optimality of geodesics is solved completely through the identification of the quotient with Heisenberg group.