Geometric Function Theory on Uniformly Quasiconformally Homogeneous Domains

Uniformly quasiconformally homogeneous domains in ${\mathbb{R}}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\ge 1.$ In this paper, we study two questions in this setting. The first is to show that quasiconformality and quasisymmetry with respect to the quasihyperbolic metric are equivalent. The second is to study normal quasiregular maps from such a domain into $S^n$ or ${\mathbb{R}}^n$ and show they enjoy geometric properties such as a uniform Hölder condition.