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

Nakamura, Gou; Nakanishi, Toshihiro

doi:10.1090/ecgd/289

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

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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