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Bers slices in families of univalent maps

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Abstract

We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions \(\varSigma \). This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial \(z\mapsto \overline{z}^d\) is the Schwarz reflection map arising from the corresponding map in \(\varSigma \). We characterize the image of this embedding in \(\varSigma \) as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial.

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Acknowledgements

The third author was supported by an endowment from Infosys Foundation and SERB research grant SRG/2020/000018.

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Authors and Affiliations

  1. University of North Texas, Denton, TX, 76203, USA

    Kirill Lazebnik

  2. California Institute of Technology, Pasadena, CA, 91106, USA

    Nikolai G. Makarov

  3. Tata Institute of Fundamental Research, Mumbai, 400005, India

    Sabyasachi Mukherjee

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  1. Kirill Lazebnik
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  2. Nikolai G. Makarov
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  3. Sabyasachi Mukherjee
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Correspondence to Kirill Lazebnik.

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Lazebnik, K., Makarov, N.G. & Mukherjee, S. Bers slices in families of univalent maps. Math. Z. 300, 2771–2808 (2022). https://doi.org/10.1007/s00209-021-02871-y

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