On Square Roots of Meromorphic Maps

Abstract

Let S be a connected Riemann surface and let \(\varphi :S \rightarrow \widehat{{\mathbb {C}}}\) be a surjective meromorphic map. A simple geometrical necessary and sufficient condition is provided for the existence of a square root of \(\varphi \), that is, a meromorphic map \(\psi :S \rightarrow \widehat{{\mathbb {C}}}\) such that \(\varphi =\psi ^{2}\).

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Acknowledgements

The first author is very grateful to Ernesto Girondo, Gabino González-Diez and Gareth Jones for the many discussions about the concept of Z-orientability. We also thanks the anonymous referee for her/his comments respect to the first version.

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Correspondence to Rubén A. Hidalgo.

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Partially supported by Projects Fondecyt 1190001 and Anillo ACT 1415 PIA-CONICYT.

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García, J.C., Hidalgo, R.A. On Square Roots of Meromorphic Maps. Results Math 74, 118 (2019). https://doi.org/10.1007/s00025-019-1042-7

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Keywords

  • Kleinian groups
  • Riemann surfaces

Mathematics Subject Classification

  • Primary 30F40
  • 30F10
  • 57M12