Elements of geometric measure theory on sub-riemannian groups

The main purpose of this thesis is to extend methods and results of geometric measure theory to the geometries of sub-riemannian groups. Typical features of sub-riemannian structures historically appeared in several fields of mathematics. Perhaps, the first seeds can be found in the 1909 work by Carathéodory on the second principle of thermodynamics. The Carathéodory theorem can be generalized to distributions of any codimension, whose Lie algebra generates the tangent space at each point. The condition on the distribution is known in Nonholonomic Mechanics, subelliptic PDE's and Optimal Control Theory as total nonholonomicity, Hormander condition, bracket generating condition or Chow condition.