Abstract
Teichmüller’s theorem gives necessary and sufficient conditions for mapping one ordered quadruple by aK-quasiconformal map onto a second ordered quadruple. We give a simple non-computational proof of the necessity part. We then characterize such extremal mappings, and obtain as a consequence a new formula for the modular function, with leads to a very simple derivation of the known expression for the Poincaré metric on the thrice-punctured sphere.
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Research partially supported by NSF grant MCS 76-04969A01.
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Kra, I. On Teichmüller’s theorem on the quasi-invariance of cross ratios. Israel J. Math. 30, 152–158 (1978). https://doi.org/10.1007/BF02760836
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DOI: https://doi.org/10.1007/BF02760836