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I wish to thank the Magnus Ehrnrooth foundation for financial support.
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Tukia, P. Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group. Acta Math. 154, 153–193 (1985). https://doi.org/10.1007/BF02392471
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DOI: https://doi.org/10.1007/BF02392471