We consider decomposition spaces R^3/G that admit defining sequences, consisting of cubes-with-handles, of finite type. Examples of such spaces are quotient spaces associated with the Whitehead continuum or with Antoine's necklaces. Semmes metrics on these spaces promote controlled topology to controlled geometry, thus providing a natural environment for analysis. We will discuss the quasisymmetric parametrizability of the Semmes spaces (R^3/G) x R^m by the Euclidean space R^{3+m}, and the quasiregular ellipticity of Semmes spaces R^3/G. There are more unanswered questions than known results. The talk is based on work with Juha Heinonen and with Pekka Pankka.
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