## Parametrizations of Teichmüller spaces by trace functions and action of mapping class groups

HTML articles powered by AMS MathViewer

- by Gou Nakamura and Toshihiro Nakanishi
- Conform. Geom. Dyn.
**20**(2016), 25-42 - DOI: https://doi.org/10.1090/ecgd/289
- Published electronically: March 18, 2016
- PDF | Request permission

## Abstract:

We give a set of trace functions which give a global parametrization of the Teichmüller space $\mathcal {T}(g,n)(L_1,\dots ,L_n)$ of hyperbolic surfaces of genus $g$ with $n$ geodesic boundary components of lengths $L_1$,…, $L_n$ such that the action of the mapping class group on the Teichmüller space can be represented by rational transformations in the parameters.## References

- Alan F. Beardon,
*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777**, DOI 10.1007/978-1-4612-1146-4 - Marc Culler,
*Lifting representations to covering groups*, Adv. in Math.**59**(1986), no. 1, 64–70. MR**825087**, DOI 10.1016/0001-8708(86)90037-X - J. Gilman and B. Maskit,
*An algorithm for $2$-generator Fuchsian groups*, Michigan Math. J.**38**(1991), no. 1, 13–32. MR**1091506**, DOI 10.1307/mmj/1029004258 - Linda Keen,
*On Fricke moduli*, 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. 205–224. MR**0288252** - Irwin Kra,
*On lifting Kleinian groups to $\textrm {SL}(2,\textbf {C})$*, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 181–193. MR**780044** - Feng Luo,
*Geodesic length functions and Teichmüller spaces*, J. Differential Geom.**48**(1998), no. 2, 275–317. MR**1630186** - Colin Maclachlan and Alan W. Reid,
*The arithmetic of hyperbolic 3-manifolds*, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR**1937957**, DOI 10.1007/978-1-4757-6720-9 - Gou Nakamura and Toshihiro Nakanishi,
*Parametrizations of some Teichmüller spaces by trace functions*, Conform. Geom. Dyn.**17**(2013), 47–57. MR**3037875**, DOI 10.1090/S1088-4173-2013-00254-3 - Toshihiro Nakanishi and Marjatta Näätänen,
*Parametrization of Teichmüller space by length parameters*, Analysis and topology, World Sci. Publ., River Edge, NJ, 1998, pp. 541–560. MR**1667832** - Toshihiro Nakanishi and Marjatta Näätänen,
*Areas of two-dimensional moduli spaces*, Proc. Amer. Math. Soc.**129**(2001), no. 11, 3241–3252. MR**1844999**, DOI 10.1090/S0002-9939-01-06010-5 - Yoshihide Okumura,
*On the global real analytic coordinates for Teichmüller spaces*, J. Math. Soc. Japan**42**(1990), no. 1, 91–101. MR**1027542**, DOI 10.2969/jmsj/04210091 - Yoshihide Okumura,
*Global real analytic length parameters for Teichmüller spaces*, Hiroshima Math. J.**26**(1996), no. 1, 165–179. MR**1380431** - R. C. Penner,
*The decorated Teichmüller space of punctured surfaces*, Comm. Math. Phys.**113**(1987), no. 2, 299–339. MR**919235**, DOI 10.1007/BF01223515 - Norman Purzitsky,
*Two-generator discrete free products*, Math. Z.**126**(1972), 209–223. MR**346070**, DOI 10.1007/BF01110724 - Paul Schmutz,
*Die Parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen*, Comment. Math. Helv.**68**(1993), no. 2, 278–288 (German). MR**1214232**, DOI 10.1007/BF02565819 - Mika Seppälä and Tuomas Sorvali,
*On geometric parametrization of Teichmüller spaces*, Ann. Acad. Sci. Fenn. Ser. A I Math.**10**(1985), 515–526. MR**802516**, DOI 10.5186/aasfm.1985.1058 - Mika Seppälä and Tuomas Sorvali,
*Parametrization of Teichmüller spaces by geodesic length functions*, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 267–284. MR**955845**, DOI 10.1007/978-1-4613-9611-6_{1}8 - Mika Seppälä and Tuomas Sorvali,
*Parametrization of Möbius groups acting in a disk*, Comment. Math. Helv.**61**(1986), no. 1, 149–160. MR**847525**, DOI 10.1007/BF02621907 - Mika Seppälä and Tuomas Sorvali,
*Traces of commutators of Möbius transformations*, Math. Scand.**68**(1991), no. 1, 53–58. MR**1124819**, DOI 10.7146/math.scand.a-12345 - Mika Seppälä and Tuomas Sorvali,
*Geometry of Riemann surfaces and Teichmüller spaces*, North-Holland Mathematics Studies, vol. 169, North-Holland Publishing Co., Amsterdam, 1992. MR**1202043** - Scott A. Wolpert,
*Geodesic length functions and the Nielsen problem*, J. Differential Geom.**25**(1987), no. 2, 275–296. MR**880186** - Heiner Zieschang,
*Finite groups of mapping classes of surfaces*, Lecture Notes in Mathematics, vol. 875, Springer-Verlag, Berlin, 1981. MR**643627**, DOI 10.1007/BFb0090465

## Bibliographic Information

**Gou Nakamura**- Affiliation: Science Division, Center for General Education, Aichi Institute of Technology,1247 Yachigusa, Yakusa, Toyota, 470-0392, Japan
- MR Author ID: 639802
- Email: gou@aitech.ac.jp
**Toshihiro Nakanishi**- Affiliation: Department of Mathematics, Shimane University, Matsue, 690-8504, Japan
- MR Author ID: 225488
- Email: tosihiro@riko.shimane-u.ac.jp
- Received by editor(s): July 2, 2015
- Received by editor(s) in revised form: January 11, 2016
- Published electronically: March 18, 2016
- Additional Notes: The first author was partially supported by the JSPS KAKENHI Grant No. 25400147.

The second author was partially supported by the JSPS KAKENHI Grant No. 22540191. - © Copyright 2016 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**20**(2016), 25-42 - MSC (2010): Primary 32G15; Secondary 30F35
- DOI: https://doi.org/10.1090/ecgd/289
- MathSciNet review: 3475293

Dedicated: Dedicated to the memory of Professor Mika Seppälä