## Teichmüller spaces of generalized symmetric homeomorphisms

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Huaying Wei and Katsuhiko Matsuzaki
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**7**(2020), 52-66

## Abstract:

We introduce the concept of a new kind of symmetric homeomorphism on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we show that the Bers Schwarzian derivative map is a holomorphic split submersion and endow a complex Banach manifold structure on the Teichmüller space of those generalized symmetric homeomorphisms.## References

- Lars V. Ahlfors,
*Quasiconformal reflections*, Acta Math.**109**(1963), 291–301. MR**154978**, DOI 10.1007/BF02391816 - Lars V. Ahlfors,
*Lectures on quasiconformal mappings*, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR**0200442** - Kari Astala and Michel Zinsmeister,
*Teichmüller spaces and BMOA*, Math. Ann.**289**(1991), no. 4, 613–625. MR**1103039**, DOI 10.1007/BF01446592 - A. Beurling and L. Ahlfors,
*The boundary correspondence under quasiconformal mappings*, Acta Math.**96**(1956), 125–142. MR**86869**, DOI 10.1007/BF02392360 - Lennart Carleson,
*On mappings, conformal at the boundary*, J. Analyse Math.**19**(1967), 1–13. MR**215986**, DOI 10.1007/BF02788706 - Guizhen Cui,
*Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces*, Sci. China Ser. A**43**(2000), no. 3, 267–279. MR**1766456**, DOI 10.1007/BF02897849 - Adrien Douady and Clifford J. Earle,
*Conformally natural extension of homeomorphisms of the circle*, Acta Math.**157**(1986), no. 1-2, 23–48. MR**857678**, DOI 10.1007/BF02392590 - Clifford J. Earle, Frederick P. Gardiner, and Nikola Lakic,
*Asymptotic Teichmüller space. II. The metric structure*, In the tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 187–219. MR**2145063**, DOI 10.1090/conm/355/06452 - Clifford J. Earle, Vladimir Markovic, and Dragomir Saric,
*Barycentric extension and the Bers embedding for asymptotic Teichmüller space*, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 87–105. MR**1940165**, DOI 10.1090/conm/311/05448 - Richard Fehlmann,
*Über extremale quasikonforme Abbildungen*, Comment. Math. Helv.**56**(1981), no. 4, 558–580 (German). MR**656212**, DOI 10.1007/BF02566227 - Frederick P. Gardiner and Nikola Lakic,
*Quasiconformal Teichmüller theory*, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR**1730906**, DOI 10.1090/surv/076 - Frederick P. Gardiner and Dennis P. Sullivan,
*Symmetric structures on a closed curve*, Amer. J. Math.**114**(1992), no. 4, 683–736. MR**1175689**, DOI 10.2307/2374795 - Hu Yun, Wu Li, and Shen Yuliang,
*On symmetric homeomorphisms on the real line*, Proc. Amer. Math. Soc.**146**(2018), no. 10, 4255–4263. MR**3834655**, DOI 10.1090/proc/14018 - Irwin Kra,
*On Teichmüller’s theorem on the quasi-invariance of cross ratios*, Israel J. Math.**30**(1978), no. 1-2, 152–158. MR**508260**, DOI 10.1007/BF02760836 - Olli Lehto,
*Univalent functions and Teichmüller spaces*, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR**867407**, DOI 10.1007/978-1-4613-8652-0 - Katsuhiko Matsuzaki,
*The universal Teichmüller space and diffeomorphisms of the circle with Hölder continuous derivatives*, Handbook of group actions. Vol. I, Adv. Lect. Math. (ALM), vol. 31, Int. Press, Somerville, MA, 2015, pp. 333–372. MR**3380337** - Subhashis Nag,
*The complex analytic theory of Teichmüller spaces*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR**927291** - Stephen W. Semmes,
*Quasiconformal mappings and chord-arc curves*, Trans. Amer. Math. Soc.**306**(1988), no. 1, 233–263. MR**927689**, DOI 10.1090/S0002-9947-1988-0927689-1 - Yuliang Shen and Huaying Wei,
*Universal Teichmüller space and BMO*, Adv. Math.**234**(2013), 129–148. MR**3003927**, DOI 10.1016/j.aim.2012.10.017 - Leon A. Takhtajan and Lee-Peng Teo,
*Weil-Petersson metric on the universal Teichmüller space*, Mem. Amer. Math. Soc.**183**(2006), no. 861, viii+119. MR**2251887**, DOI 10.1090/memo/0861 - H. Wei and K. Matsuzaki,
*Symmetric and strongly symmetric homeomorphisms on the real line with non-symmetric inversion*, arXiv:1905.05933, 2019.

## Additional Information

**Huaying Wei**- Affiliation: Department of Mathematics and Statistics, Jiangsu Normal University, Xuzhou221116, People’s Republic of China
- Email: hywei@jsnu.edu.cn
**Katsuhiko Matsuzaki**- Affiliation: Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo 169-8050, Japan
- MR Author ID: 294335
- ORCID: 0000-0003-0025-5372
- Email: matsuzak@waseda.jp
- Received by editor(s): May 29, 2019
- Received by editor(s) in revised form: February 13, 2020
- Published electronically: May 15, 2020
- Additional Notes: This research was supported by the National Natural Science Foundation of China (Grant No. 11501259) and Japan Society for the Promotion of Science (KAKENHI 18H01125).
- Communicated by: Jeremy Tyson
- © Copyright 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B
**7**(2020), 52-66 - MSC (2010): Primary 30F60, 30C62, 32G15; Secondary 37E10, 58D05
- DOI: https://doi.org/10.1090/bproc/47
- MathSciNet review: 4098591