Abstract
Let \(S_0\) be a Riemann surface of infinite type without boundary and admitting an upper and lower bounded pair of pants decomposition, and let \({ AT}(S_0)\) be the asymptotic Teichmüller space of S. In this paper, we first give a length spectrum definition of such a Teichmüller space \({ AT}(S_0)\). Then we introduce a length spectrum metric on \({ AT}(S_0)\) and show that it induces the same topology on \({ AT}(S_0)\) as the Teichmüller metric. We also show that these two metrics are not bi-Lipschitz.
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Acknowledgements
Both authors are grateful to the referee for providing corrections of typos and very useful comments, especially a short Proof of Proposition 3.
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Communicated by Adrian Constantin.
J. Hu: Research partially supported by PSC-CUNY research awards. F. G. Jimenez-Lopez: Research partially supported by CONACyT posdoctoral fellowship.
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Hu, J., Jimenez-Lopez, F.G. Length spectrum characterization of asymptotic Teichmüller space. Monatsh Math 186, 73–91 (2018). https://doi.org/10.1007/s00605-018-1176-9
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DOI: https://doi.org/10.1007/s00605-018-1176-9
Keywords
- Riemann surface
- Teichmüller space
- Asymptotic Teichmüller space
- Teichmüller metric
- Length spectrum metric