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Koebe Spaces of Infinite Type

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Abstract

Let D be a Koebe circle domain in the complex plane of infinite type. We show that, when D is tame, D corresponds to an infinitely generated extended classical Schottky group G and the quasiconformal Koebe space of D can be identified with the quasiconformal deformation space of G. Moreover, when D also satisfies the boundedness conditions, we prove that the essential Teichmüller space of the G-orbit of \(\infty \) admits the complex analytic structure.

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References

  1. Fujikawa, E., Taniguchi, M.: The Teichmüller space of a countable set of points on a Riemann surface. Conf. Geom. Dyn. 21, 64–77 (2017)

    Article  MATH  Google Scholar 

  2. He, Z., Schramm, O.: Fixed points, Koebe uniformization and circle packing. Ann. Math. 137, 369–406 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Koebe, P.: Abhandlungen zur Theorie der Konformen Abbildung; IV. Math. Z. 7, 235–301 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, GMW, vol. 126. Springer, New York (1973)

    Book  MATH  Google Scholar 

  5. Matsuzaki, K.: Infinite-dimensional Teichmüller spaces and modular groups. Handbook of Teichmüller theory IV IRMA-LNMP, EMS, vol. 19, pp. 681–716 (2014)

  6. Matsuzaki, K., Taniguchi, M.: Hyperbolic Manifolds and Kleinian Groups. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  7. McMullen, C.: Complex dynamics and renormalization. Ann. Math. Studies, vol. 135. Princeton (1994)

  8. Taniguchi, M., Maitani, F.: A condition for an infinitely generated Schottky group to be classical. Annual Rep. Graduate School, vol. 27, pp. 181–188. Nara Women’s University (2012)

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Acknowledgements

The author would like to express heartfelt thanks to the referee for his/her helpful comments.

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Authors and Affiliations

  1. Nara Women’s University, Nara, 630-8506, Japan

    Masahiko Taniguchi

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  1. Masahiko Taniguchi
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Correspondence to Masahiko Taniguchi.

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Communicated by Kenneth Stephenson.

The author was partially supported by Grants-in-Aid for Scientific Research (C) Grant no. 15K04925.

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Taniguchi, M. Koebe Spaces of Infinite Type. Comput. Methods Funct. Theory 18, 537–544 (2018). https://doi.org/10.1007/s40315-018-0235-5

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  • DOI: https://doi.org/10.1007/s40315-018-0235-5

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