Pure mapping class group acting on Teichmüller space

Fujikawa, Ege

doi:10.1090/S1088-4173-08-00188-4

Pure mapping class group acting on Teichmüller space
HTML articles powered by AMS MathViewer

by Ege Fujikawa

Conform. Geom. Dyn. 12 (2008), 227-239

DOI: https://doi.org/10.1090/S1088-4173-08-00188-4

Published electronically: December 23, 2008

PDF | Request permission

Abstract:

For a Riemann surface of analytically infinite type, the action of the quasiconformal mapping class group on the Teichmüller space is not discontinuous in general. In this paper, we consider pure mapping classes that fix all topological ends of a Riemann surface and prove that the pure mapping class group acts on the Teichmüller space discontinuously under a certain geometric condition of a Riemann surface. We also consider the action of the quasiconformal mapping class group on the asymptotic Teichmüller space. Non-trivial mapping classes can act on the asymptotic Teichmüller space trivially. We prove that all such mapping classes are contained in the pure mapping class group.

References

Joan S. Birman, Mapping class groups of surfaces, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 13–43. MR 975076, DOI 10.1090/conm/078/975076
Corneliu Constantinescu and Aurel Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 32, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 (German). MR 0159935, DOI 10.1007/978-3-642-87031-6
Clifford J. Earle and Frederick P. Gardiner, Geometric isomorphisms between infinite-dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1163–1190. MR 1322950, DOI 10.1090/S0002-9947-96-01490-0
C. J. Earle, F. P. Gardiner and N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, I.H.E.S., 1995, preprint.
C. J. Earle, F. P. Gardiner, and N. Lakic, Asymptotic Teichmüller space. I. The complex structure, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 17–38. MR 1759668, DOI 10.1090/conm/256/03995
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
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
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, Modular groups acting on infinite dimensional Teichmüller spaces, In the tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 239–253. MR 2145066, DOI 10.1090/conm/355/06455
Ege Fujikawa, The order of periodic elements of Teichmüller modular groups, Tohoku Math. J. (2) 57 (2005), no. 1, 45–51. MR 2113989
Ege Fujikawa, The action of geometric automorphisms of asymptotic Teichmüller spaces, Michigan Math. J. 54 (2006), no. 2, 269–282. MR 2252759, DOI 10.1307/mmj/1156345593
Ege Fujikawa, Another approach to the automorphism theorem for Teichmüller spaces, In the tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 39–44. MR 2342805, DOI 10.1090/conm/432/08298
E. Fujikawa, Limit set of quasiconformal mapping class group on asymptotic Teichmüller space, Proceedings of the international workshop on Teichmüller theory and moduli problems, Lecture note series in the Ramanujan Math. Soc., to appear.
Ege Fujikawa and Katsuhiko Matsuzaki, Non-stationary and discontinuous quasiconformal mapping class groups, Osaka J. Math. 44 (2007), no. 1, 173–185. MR 2313034
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
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
F. P. Gardiner and N. Lakic, A vector field approach to mapping class actions, Proc. London Math. Soc. (3) 92 (2006), no. 2, 403–427. MR 2205723, DOI 10.1112/S0024611505015558
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
Dennis Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 165–179. MR 718141, DOI 10.1090/conm/020/718141
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
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, Inclusion relations between the Bers embeddings of Teichmüller spaces, Israel J. Math. 140 (2004), 113–123. MR 2054840, DOI 10.1007/BF02786628
Katsuhiko Matsuzaki, A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2573–2579. MR 2302578, DOI 10.1090/S0002-9939-07-08761-8
Hideki Miyachi, A reduction for asymptotic Teichmüller spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 55–71. MR 2297877
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
L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. MR 0264064, DOI 10.1007/978-3-642-48269-4
Masahiko Taniguchi, The Teichmüller space of the ideal boundary, Hiroshima Math. J. 36 (2006), no. 1, 39–48. MR 2213642