Fujikawa, Ege Pure mapping class group acting on Teichmüller space. (English) Zbl 1195.30066 Conform. Geom. Dyn. 12, 227-239 (2008). Given a Riemann surface \(R\), the quasiconformal mapping class group \(MCG(R)\) is the set of all homotopy equivalence classes of its quasiconformal automorphisms. If \(R\) has finite type, \(MCG(R)\) acts on the Teichmüller space \(T(R)\) discontinuously. For surfaces of inifinite type, this does not in general hold.However, by passing to an appropriate subgroup, the pure mapping class group, \(P(R)\), defined as the set of elements in \(MCG(R)\) that fix all non-cuspidal ends of \(R\), one can obtain a discontinuity result. The main result of the paper is that for all \(R\) satisfying an appropriate bounded geometry condition (which is satisfied, for example, by all normal covers of a compact surface except the universal cover), \(P(R)\) does act discontinuously on \(T(R)\). This extends an earlier result of F. P. Gardiner and N. Lakic [Proc. Lond. Math. Soc., III. Ser. 92, No. 2, 403–427 (2006; Zbl 1088.30013)], who considered the case where \(R\) was the Riemann sphere with the middle-third Cantor set removed.It is also shown that \(P(R)\) coincides with the set of all mapping classes which leave invariant the homology class of all simple separating curves on the surface \(\dot{R}\), which is obtained from \(R\) by filling in the punctures. Reviewer: Jayadev Athreya (New Haven) Cited in 4 Documents MSC: 30F60 Teichmüller theory for Riemann surfaces 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) Keywords:Mapping class group; Riemann surface of infinite type; asymptotic Teichmüller space Citations:Zbl 1088.30013 PDFBibTeX XMLCite \textit{E. Fujikawa}, Conform. Geom. Dyn. 12, 227--239 (2008; Zbl 1195.30066) Full Text: DOI References: [1] 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. [2] Corneliu Constantinescu and Aurel Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 32, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 (German). · Zbl 0112.30801 [3] Clifford J. Earle and Frederick P. Gardiner, Geometric isomorphisms between infinite-dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1163 – 1190. · Zbl 0873.32020 [4] C. J. Earle, F. P. Gardiner and N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, I.H.E.S., 1995, preprint. · Zbl 0958.30033 [5] 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. · Zbl 0973.30032 [6] 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. · Zbl 1112.30034 [7] 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. · Zbl 1020.30047 [8] 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. · Zbl 0964.30026 [9] Ege Fujikawa, Limit sets and regions of discontinuity of Teichmüller modular groups, Proc. Amer. Math. Soc. 132 (2004), no. 1, 117 – 126 (electronic). · Zbl 1091.30012 [10] 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. · Zbl 1072.30032 [11] Ege Fujikawa, The order of periodic elements of Teichmüller modular groups, Tohoku Math. J. (2) 57 (2005), no. 1, 45 – 51. · Zbl 1071.30041 [12] Ege Fujikawa, The action of geometric automorphisms of asymptotic Teichmüller spaces, Michigan Math. J. 54 (2006), no. 2, 269 – 282. · Zbl 1115.30051 [13] 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. · Zbl 1189.30085 [14] 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. · Zbl 1209.30018 [15] Ege Fujikawa and Katsuhiko Matsuzaki, Non-stationary and discontinuous quasiconformal mapping class groups, Osaka J. Math. 44 (2007), no. 1, 173 – 185. · Zbl 1119.30023 [16] 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. · Zbl 1064.30047 [17] Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. · Zbl 0949.30002 [18] 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. · Zbl 1088.30013 [19] Frederick P. Gardiner and Dennis P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), no. 4, 683 – 736. · Zbl 0778.30045 [20] 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. · Zbl 0553.57002 [21] Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. · Zbl 0606.30001 [22] Vladimir Markovic, Biholomorphic maps between Teichmüller spaces, Duke Math. J. 120 (2003), no. 2, 405 – 431. · Zbl 1056.30045 [23] Katsuhiko Matsuzaki, Inclusion relations between the Bers embeddings of Teichmüller spaces, Israel J. Math. 140 (2004), 113 – 123. · Zbl 1056.30046 [24] Katsuhiko Matsuzaki, A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2573 – 2579 (electronic). · Zbl 1111.30023 [25] Hideki Miyachi, A reduction for asymptotic Teichmüller spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 55 – 71. · Zbl 1116.32007 [26] 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. · Zbl 0667.30040 [27] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603 [28] Masahiko Taniguchi, The Teichmüller space of the ideal boundary, Hiroshima Math. J. 36 (2006), no. 1, 39 – 48. · Zbl 1116.30029 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.