Transitive bilipschitz group actions and bilipschitz parametrizations David Freeman 30C6222E2551F99bilipschitz homogeneitymetric inversion We prove that Ahlfors $2$-regular quasisymmetric images of $\mathbb{R}^2$ are bi-Lipschitz images of $\mathbb{R}^2$ 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4872 10.1512/iumj.2013.62.4872 en Indiana Univ. Math. J. 62 (2013) 311 - 331 state-of-the-art mathematics http://iumj.org/access/