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It is known that every doubling metric space admits quasisymmetric map into Euclidean space. My question is, is there a known explicit (closed-form) quasisymmetry from the Heisenberg group into a Euclidean space?

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    $\begingroup$ I doubt. The geometry of the Heisenberg group is very different from the Euclidean one so in order for an embedding to be nice, it needs to have fractal nature and it is unlikely that you can find an explicit formula. Correct me if I am wrong. $\endgroup$ – Piotr Hajlasz Jan 30 at 19:54
  • $\begingroup$ I also expected as much, but thought I should still ask just in case. As always, thanks @PiotrHajlasz :) $\endgroup$ – AIM_BLB Jan 30 at 20:38

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