Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups

In any Carnot (nilpotent stratified Lie) group G of homogeneous dimension Q, Green's function u for the Q-Laplace equation exists and is unique. We prove that there exists a constant gamma = gamma(G) so that N = e(-gammau) is a homogeneous norm in G. Then the extremal lengths of spherical ring domains (measured with respect to N) can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the Carnot-Caratheodory metric. Applications include regularity properties of quasiconformal mappings and a geometric characterization of bi-Lipschitz mappings.