Geometry and quasisymmetric parametrization of Semmes spaces

We consider decomposition spaces ${\mathbb{R}}^3\mathrm{/}G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on ${\mathbb{R}}^3\mathrm{/}G$ constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space ${\mathbb{R}}^3\mathrm{/}G\times {\mathbb{R}}^m$ imposes quantitative topological constraints, in terms of the circulation and growth, to the defining sequences for ${\mathbb{R}}^3\mathrm{/}G$. We give a necessary condition and a sufficient condition for the existence of parametrization. The necessary condition answers a question of Heinonen and Semmes on quasisymmetric parametrizability of spaces associated to the Bing double. The sufficient condition gives new examples of quasispheres in ${\text{bS}}^4$.