The Nielsen realization problem for asymptotic Teichmüller modular groups

Fujikawa, Ege; Matsuzaki, Katsuhiko

doi:10.1090/S0002-9947-2013-05767-4

The Nielsen realization problem for asymptotic Teichmüller modular groups
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by Ege Fujikawa and Katsuhiko Matsuzaki PDF

Trans. Amer. Math. Soc. 365 (2013), 3309-3327 Request permission

Abstract:

Under a certain geometric assumption on a hyperbolic Riemann surface, we prove an asymptotic version of the fixed point theorem for the Teichmüller modular group, which asserts that every finite subgroup of the asymptotic Teichmüller modular group has a common fixed point in the asymptotic Teichmüller space. For its proof, we use a topological characterization of the asymptotically trivial mapping class group, which has been obtained in the authors’ previous paper, but a simpler argument is given here. As a consequence, every finite subgroup of the asymptotic Teichmüller modular group is realized as a group of quasiconformal automorphisms modulo coincidence near infinity. Furthermore, every finite subgroup of a certain geometric automorphism group of the asymptotic Teichmüller space is realized as an automorphism group of the Royden boundary of the Riemann surface. These results can be regarded as asymptotic versions of the Nielsen realization theorem.

References

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C. J. Earle, F. P. Gardiner and N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, I.H.E.S., 1995, unpublished manuscript.
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
Richard Fehlmann, Über extremale quasikonforme Abbildungen, Comment. Math. Helv. 56 (1981), no. 4, 558–580 (German). MR 656212, DOI 10.1007/BF02566227
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, Pure mapping class group acting on Teichmüller space, Conform. Geom. Dyn. 12 (2008), 227–239. MR 2466018, DOI 10.1090/S1088-4173-08-00188-4
Ege Fujikawa, Limit set of quasiconformal mapping class group on asymptotic Teichmüller space, Teichmüller theory and moduli problem, Ramanujan Math. Soc. Lect. Notes Ser., vol. 10, Ramanujan Math. Soc., Mysore, 2010, pp. 167–178. MR 2667555
Ege Fujikawa and Katsuhiko Matsuzaki, Stable quasiconformal mapping class groups and asymptotic Teichmüller spaces, Amer. J. Math. 133 (2011), no. 3, 637–675. MR 2808328, DOI 10.1353/ajm.2011.0017
Ege Fujikawa, Katsuhiko Matsuzaki, and Masahiko Taniguchi, Structure theorem for holomorphic self-covers of Riemann surfaces and its applications, Infinite dimensional Teichmüller spaces and moduli spaces, RIMS Kôkyûroku Bessatsu, B17, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, pp. 21–36. MR 2560682
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
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
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
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
Fumio Maitani, On rigidity of an end under conformal mappings preserving the infinite homology basis, Complex Variables Theory Appl. 24 (1994), no. 3-4, 281–287. MR 1270318, DOI 10.1080/17476939408814720
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
Vladimir Markovic, Quasisymmetric groups, J. Amer. Math. Soc. 19 (2006), no. 3, 673–715. MR 2220103, DOI 10.1090/S0894-0347-06-00518-2
Vladimir Markovic, Realization of the mapping class group by homeomorphisms, Invent. Math. 168 (2007), no. 3, 523–566. MR 2299561, DOI 10.1007/s00222-007-0039-0
V. Markovic and D. Šarić, The mapping class group cannot be realized by homeomorphisms, preprint.
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 countable Teichmüller modular group, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3119–3131. MR 2135738, DOI 10.1090/S0002-9947-04-03765-1
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
Katsuhiko Matsuzaki, The action of elliptic modular transformations on asymptotic Teichmüller spaces, Teichmüller theory and moduli problem, Ramanujan Math. Soc. Lect. Notes Ser., vol. 10, Ramanujan Math. Soc., Mysore, 2010, pp. 481–488. MR 2667568
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
Mitsuru Nakai, Existence of quasiconformal mappings between Riemann surfaces, Hokkaido Math. J. 10 (1981), no. Special Issue, 525–530. MR 662318
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