Rectifiable sets and intrinsic Lipschitz graphs within Carnot Groups

A Carnot group is a connected, simply connected nilpotent Lie group whose Lie algebra is stratified. In these manifolds naturally arises a distance, called the Carnot-Caratheodory metric, that makes them sub-Riemannian manifolds. We define in Carnot groups homogeneous subgroups, complementary subgroups, intrinsic graphs, intrinsic Lipschitz graphs intrinsic differentiability and uniform intrinsic differentiability. We build two definitions of regular submanifolds in Carnot groups, that generalize the previous notions of H-regular surfaces and hypersurfaces. Then, we give two consequent notions of intrinsic rectifiable sets in Carnot groups and we prove their equivalence in groups of type *, as a consequence of a Rademacher's type, theorem for one codimensional intrinsic graphs.