## Some metrics on Teichmüller spaces of surfaces of infinite type

HTML articles powered by AMS MathViewer

- by Lixin Liu and Athanase Papadopoulos PDF
- Trans. Amer. Math. Soc.
**363**(2011), 4109-4134 Request permission

## Abstract:

Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topologically infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichmüller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichmüller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichmüller theory of surfaces of infinite topological type that do not appear in the setting of the Teichmüller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichmüller spaces associated to a given surface of topologically infinite type.## References

- William Abikoff,
*The real analytic theory of Teichmüller space*, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR**590044** - 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** - D. Alessandrini, L. Liu, A. Papadopoulos and W. Su, On various Teichmüller spaces of a surface of infinite topological type, to appear in the Proceedings of the Amer. Math. Soc.
- Peter Buser,
*Geometry and spectra of compact Riemann surfaces*, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR**1183224** - Ara Basmajian,
*Hyperbolic structures for surfaces of infinite type*, Trans. Amer. Math. Soc.**336**(1993), no. 1, 421–444. MR**1087051**, DOI 10.1090/S0002-9947-1993-1087051-2 - Mikhael Gromov,
*Structures métriques pour les variétés riemanniennes*, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR**682063** - 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 and Irwin Kra,
*On holomorphic mappings between Teichmüller spaces*, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 107–124. MR**0430319** - C. Earle, F. Gardiner & N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, Preprint, IHES, 1995.
- Adam Lawrence Epstein,
*Effectiveness of Teichmüller modular groups*, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 69–74. MR**1759670**, DOI 10.1090/conm/256/03997 *Travaux de Thurston sur les surfaces*, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR**568308**- A. Fletcher,
*Local rigidity of infinite-dimensional Teichmüller spaces*, J. London Math. Soc. (2)**74**(2006), no. 1, 26–40. MR**2254550**, DOI 10.1112/S0024610706023003 - Ege Fujikawa,
*Limit sets and regions of discontinuity of Teichmüller modular groups*, Proc. Amer. Math. Soc.**132**(2004), no. 1, 117–126. MR**2021254**, DOI 10.1090/S0002-9939-03-06988-0 - Ege Fujikawa, Hiroshige Shiga, and Masahiko Taniguchi,
*On the action of the mapping class group for Riemann surfaces of infinite type*, J. Math. Soc. Japan**56**(2004), no. 4, 1069–1086. MR**2091417**, DOI 10.2969/jmsj/1190905449 - Y. Imayoshi and M. Taniguchi,
*An introduction to Teichmüller spaces*, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR**1215481**, DOI 10.1007/978-4-431-68174-8 - B. V. Kerékjártó, Vorlesungen über Topologie. I, Varlag von Julius Springer, Berlin, 1923.
- Nikola Lakic,
*An isometry theorem for quadratic differentials on Riemann surfaces of finite genus*, Trans. Amer. Math. Soc.**349**(1997), no. 7, 2951–2967. MR**1390043**, DOI 10.1090/S0002-9947-97-01771-6 - Zhong Li,
*Teichmüller metric and length spectrums of Riemann surfaces*, Sci. Sinica Ser. A**29**(1986), no. 3, 265–274. MR**855233** - Liu Lixin,
*On the length spectrum of non-compact Riemann surfaces*, Ann. Acad. Sci. Fenn. Math.**24**(1999), no. 1, 11–22. MR**1678001** - Li Xin Liu,
*The metrics of length spectrum in Teichmüller space*, Chinese Ann. Math. Ser. A**22**(2001), no. 1, 19–26 (Chinese, with Chinese summary); English transl., Chinese J. Contemp. Math.**22**(2001), no. 1, 23–34. MR**1826852** - Lixin Liu, Zongliang Sun, and Hanbai Wei,
*Topological equivalence of metrics in Teichmüller space*, Ann. Acad. Sci. Fenn. Math.**33**(2008), no. 1, 159–170. MR**2386845** - Vladimir Markovic,
*Biholomorphic maps between Teichmüller spaces*, Duke Math. J.**120**(2003), no. 2, 405–431. MR**2019982**, DOI 10.1215/S0012-7094-03-12028-1 - Katsuhiko Matsuzaki,
*The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces*, Kodai Math. J.**26**(2003), no. 3, 279–287. MR**2018722**, DOI 10.2996/kmj/1073670609 - Katsuhiko Matsuzaki,
*Inclusion relations between the Bers embeddings of Teichmüller spaces*, Israel J. Math.**140**(2004), 113–123. MR**2054840**, DOI 10.1007/BF02786628 - 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** - A. O. Prishlyak and K. I. Mischenko,
*Classification of noncompact surfaces with boundary*, Methods Funct. Anal. Topology**13**(2007), no. 1, 62–66. MR**2308580** - Ian Richards,
*On the classification of noncompact surfaces*, Trans. Amer. Math. Soc.**106**(1963), 259–269. MR**143186**, DOI 10.1090/S0002-9947-1963-0143186-0 - H. L. Royden,
*Automorphisms and isometries of Teichmüller space*, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 369–383. MR**0288254** - Hiroshige Shiga,
*On a distance defined by the length spectrum of Teichmüller space*, Ann. Acad. Sci. Fenn. Math.**28**(2003), no. 2, 315–326. MR**1996441** - William P. Thurston,
*Three-dimensional geometry and topology. Vol. 1*, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR**1435975**

## Additional Information

**Lixin Liu**- Affiliation: Department of Mathematics, Sun Yat-sen (Zongshan) University, 510275, Guangzhou, People’s Republic of China
- Email: mcsllx@mail.sysu.edu.cn
**Athanase Papadopoulos**- Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- MR Author ID: 135835
- Email: papadopoulos@math.u-strasbg.fr
- Received by editor(s): June 23, 2008
- Received by editor(s) in revised form: March 16, 2009, and April 18, 2009
- Published electronically: March 23, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**363**(2011), 4109-4134 - MSC (2000): Primary 32G15, 30F30, 30F60
- DOI: https://doi.org/10.1090/S0002-9947-2011-05090-7
- MathSciNet review: 2792982