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Stars at infinity for boundaries of Teichmüller space

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Abstract

In this paper, we study the stars in the Thurston boundary and the Teichmüller boundary of Teichmüller space. Precisely, we consider a conjecture posed by Karlsson: with respect to the Teichmüller metric on Teichmüller space, the star of a boundary point in the Thurston boundary is equal to its zero set. This conjecture was partially solved by Karlsson and Duchin-Fisher. For the unsolved part of this conjecture, we obtain a new result which improves Karlsson’s result. And we prove two modified versions of this conjecture. Firstly, we prove that this conjecture is true if we replace the Thurston boundary by the Teichmüller boundary, that is, similarly defining the zero set of a boundary point in the Teichmüller boundary, we prove that the star of a boundary point in the Teichmüller boundary is equal to its zero set. Secondly, we prove that this conjecture is true if we replace Teichmüller metric by Thurston’s asymmetric metric, that is, similarly defining the star of a boundary point in the Thurston boundary for Thurston’s asymmetric metric, we prove that the star of a boundary point with respect to Thurston’s asymmetric metric is equal to its zero set.

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Acknowledgements

The authors would like to thank Professor Lixin Liu for many useful suggestions and discussions.

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Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, People’s Republic of China

    Peijia Liu

  2. School of Mathematics and Computational Science, Wuyi University, Jiangmen, 529020, Guangdong, People’s Republic of China

    Yaozhong Shi

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  1. Peijia Liu
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  2. Yaozhong Shi
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Correspondence to Yaozhong Shi.

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The work was partially supported by NSFC, No: 12271533.

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Liu, P., Shi, Y. Stars at infinity for boundaries of Teichmüller space. Geom Dedicata 218, 8 (2024). https://doi.org/10.1007/s10711-023-00853-4

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