Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


The Nielsen realization problem for asymptotic Teichmüller modular groups

Authors: Ege Fujikawa and Katsuhiko Matsuzaki
Journal: Trans. Amer. Math. Soc. 365 (2013), 3309-3327
MSC (2010): Primary 30F60, 32G15; Secondary 37F30, 30F25
Published electronically: January 4, 2013
MathSciNet review: 3034467
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. C. J. Earle and F. P. Gardiner, Geometric isomorphisms between infinite dimensional Teichmüller spaces, Trans. Amer. Math. Soc. 348 (1996), 1163-1190. MR 1322950 (96h:32024)
  • 2. C. J. Earle, F. P. Gardiner and N. Lakic, Teichmüller spaces with asymptotic conformal equivalence, I.H.E.S., 1995, unpublished manuscript.
  • 3. C. J. Earle, F. P. Gardiner and N. Lakic, Asymptotic Teichmüller space, Part I: The complex structure, In the tradition of Ahlfors and Bers, Contemporary Math. 256 (2000), 17-38. MR 1759668 (2001m:32029)
  • 4. C. J. Earle, F. P. Gardiner and N. Lakic, Asymptotic Teichmüller space, Part II: The metric structure, In the tradition of Ahlfors and Bers, III, Contemporary Math. 355 (2004), 187-219. MR 2145063 (2006g:30078)
  • 5. C. J. Earle, V. Markovic and D. Šarić, Barycentric extension and the Bers embedding for asymptotic Teichmüller space, Complex manifolds and hyperbolic geometry, Contemporary Math. 311 (2002), 87-105. MR 1940165 (2003i:30072)
  • 6. A. L. Epstein, Effectiveness of Teichmüller modular groups, In the tradition of Ahlfors and Bers, Contemporary Math. 256 (2000), 69-74. MR 1759670 (2001a:30059)
  • 7. R. Fehlmann, Über extremale quasikonforme Abbildungen, Comment Math. Helv. 56 (1981), 558-580. MR 656212 (83e:30024)
  • 8. E. Fujikawa, The order of periodic elements of Teichmüller modular groups, Tohoku Math. J. 57 (2005), 45-51. MR 2113989 (2005j:30059)
  • 9. E. Fujikawa, Pure mapping class group acting on Teichmüller space, Conform. Geom. Dyn. 12 (2008), 227-239. MR 2466018 (2009j:30103)
  • 10. E. Fujikawa, Limit set of quasiconformal mapping class group on asymptotic Teichmüller space, Teichmüller theory and moduli problems, Lecture note series in the Ramanujan Math. Soc. 10 (2009), 167-178. MR 2667555 (2011g:30097)
  • 11. E. Fujikawa and K. Matsuzaki, Stable quasiconformal mapping class groups and asymptotic Teichmüller spaces, Amer. J. Math. 133 (2011), 637-675. MR 2808328
  • 12. E. Fujikawa, K. Matsuzaki and M. Taniguchi, Structure theorem for holomorphic self-covers and its applications, Infinite dimensional Teichmüller space and moduli space, RIMS Kôkyûroku Bessatsu, Research Institute for Mathematical Sciences, Kyoto University, B17 (2010), 21-36. MR 2560682 (2012b:30085)
  • 13. F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs 76, Amer. Math. Soc., 2000. MR 1730906 (2001d:32016)
  • 14. F. P. Gardiner and D. P. Sullivan, Symmetric structure on a closed curve, Amer. J. Math. 114 (1992), 683-736. MR 1175689 (95h:30020)
  • 15. S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), 235-265. MR 690845 (85e:32029)
  • 16. O. Lehto, Univalent Functions and Teichmüller Spaces, Graduate Texts in Math. 109, Springer, 1986. MR 867407 (88f:30073)
  • 17. F. Maitani, On rigidity of an end under conformal mapping preserving the infinity homology basis, Complex Variables 24 (1994), 281-287. MR 1270318 (95e:30010)
  • 18. V. Markovic, Biholomorphic maps between Teichmüller spaces, Duke Math. J. 120 (2003), 405-431. MR 2019982 (2004h:30058)
  • 19. V. Markovic, Quasisymmetric groups, J. Amer. Math. Soc. 19 (2006), 673-715. MR 2220103 (2007c:37057)
  • 20. V. Markovic, Realization of the mapping class group by homeomorphisms, Invent. Math. 168 (2007), 523-566. MR 2299561 (2008c:57033)
  • 21. V. Markovic and D. Šarić, The mapping class group cannot be realized by homeomorphisms, preprint.
  • 22. K. Matsuzaki, Inclusion relations between the Bers embeddings of Teichmüller spaces, Israel J. Math. 140 (2004), 113-124. MR 2054840 (2005e:30077)
  • 23. K. Matsuzaki, A countable Teichmüller modular group, Trans. Amer. Math. Soc. 357 (2005), 3119-3131. MR 2135738 (2006f:30052)
  • 24. K. Matsuzaki, A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space, Proc. Amer. Math. Soc. 135 (2007), 2573-2579. MR 2302578 (2008d:30071)
  • 25. K. Matsuzaki, The action of elliptic modular transformations on asymptotic Teichmüller spaces, Proceedings of the international workshop on Teichmüller theory and moduli problems, Lecture Note Series in the Ramanujan Mathematical Society, 10 (2010), 481-488. MR 2667568 (2011k:30054)
  • 26. S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Canadian Math. Soc. Ser. of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, 1988. MR 927291 (89f:32040)
  • 27. M. Nakai, Existence of quasiconformal mappings between Riemann surfaces, Hokkaido Math. J. 10 (1981), Special Issue, 525-530. MR 662318 (83f:30013)
  • 28. L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften 164, Springer, 1970. MR 0264064 (41:8660)
  • 29. M. Taniguchi, The Teichmüller space of the ideal boundary, Hiroshima Math. J. 36 (2006), 39-48. MR 2213642 (2007i:30065)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30F60, 32G15, 37F30, 30F25

Retrieve articles in all journals with MSC (2010): 30F60, 32G15, 37F30, 30F25

Additional Information

Ege Fujikawa
Affiliation: Department of Mathematics, Chiba University, Inage-ku, Chiba 263-8522, Japan

Katsuhiko Matsuzaki
Affiliation: Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo 169-8050, Japan

Keywords: Riemann surface, hyperbolic geometry, Teichmüller space, quasiconformal mapping class group
Received by editor(s): May 22, 2011
Received by editor(s) in revised form: November 24, 2011
Published electronically: January 4, 2013
Dedicated: Dedicated to Professor Masahiko Taniguchi on the occasion of his 60th birthday
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society