Abstract
We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.
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The authors thank Jeremy Tyson and Colleen Ackermann for comments on a draft of this paper. They also thank the referee for useful feedback.
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D. Jung was supported by U.S. Department of Education GAANN fellowship P200A150319.
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Gartland, C., Jung, D. & Romney, M. Quasiconformal mappings on the Grushin plane. Math. Z. 287, 915–928 (2017). https://doi.org/10.1007/s00209-017-1851-x
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DOI: https://doi.org/10.1007/s00209-017-1851-x