Geometry and quasisymmetric parametrization of Semmes spaces

We consider decomposition spaces R 3 /G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R 3 /G constructed via modular embeddings of R 3 /G into a Euclidean space promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R 3 /G×R m by R 3+m for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for R 3 /G. We give a necessary condition and a sufficient condition for the existence of such a parametrization. The necessary condition answers negatively 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 S 4 .