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On the properties of various compactifications of Teichmüller space

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Abstract

In this paper, we study the Gardiner–Masur compactification, the Teichmüller compactification and the asymptotic visual compactification of Teichmüller space. We prove that the Gardiner–Masur compactification is stronger than the Teichmüller compactification. We also prove that the asymptotic visual compactification is independent of the base point. Moreover, we prove that in these three compactifications, the convergences (of a sequence in Teichmüller space) to a same indecomposable measured foliation are equivalent. As an application, we construct some counter examples about the relation between the Gardiner–Masur compactification and the Thurston compactification, and the relation between the Teichmüller compactifications with different base points.

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Data Availability

The data that support the findings of this study are available on Scitation. The data that support the findings of this study are also available from the corresponding author upon reasonable request.

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Acknowledgements

The authors would like to thank the referee for the careful reading and many valuable suggestions, especially for suggesting the diagram in the proof of Proposition 4.2 and for providing the code of the diagram.

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  1. Sun Yat-sen University, School of Mathematics, 510275, Guangzhou, People’s Republic of China

    Lixin Liu & Yaozhong Shi

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

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Communicated by Andreas Cap.

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The work was partially supported by NSFC, No: 11771456.

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Liu, L., Shi, Y. On the properties of various compactifications of Teichmüller space. Monatsh Math 198, 371–391 (2022). https://doi.org/10.1007/s00605-022-01680-7

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