Quasiconformality and quasisymmetry in metric measure spaces

A homeomorphism f: X → Y between metric spaces is called quasisymmetric if it satisfies the three-point condition of Tukia and Väisälä. It has been known since the 1960's that when X = Y = Rⁿ (n ≥ 2), the class of quasisymmetric maps coincides with the class of quasiconformal maps, i.e. those homeomorphisms f: Rⁿ → Rⁿ which quasipreserve the conformal moduli of all families of curves. We prove that quasisymmetry implies quasiconformality in the case that the metric spaces in question are locally compact and connected and have Hausdorff dimension Q > 1 quantitatively. The main conceptual tool in the proof is a discrete version of the conformal modulus due to Pansu.