  We prove that Ahlfors 2-regular quasisymmetric images of the Euclidean plane
are bi-Lipschitz images of the plane if and only if they are uniformly
bi-Lipschitz homogeneous with respect to a group. We also prove that certain
geodesic spaces are bi-Lipschitz images of Carnot groups if they are inversion
invariant bi-Lipschitz homogeneous with respect to a group.
