The difficulties with sub-Riemannian geometry in infinite dimensions

Many methods of shape analysis, such as the LDDMM framework, use sub-Riemannian geometry. Moreover, one often needs to consider infinite dimensions. However, in this case, many complications can arise. I will give a brief overview of what can and cannot be done in infinite dimensional sub-riemannian manifolds, and give an in-depth study of the geodesics of a non-trivial example: the free 2-step nilpotent group modeled on a Hilbert space.