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
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Additional Information
- Ege Fujikawa
- Affiliation: Department of Mathematics, Chiba University, Inage-ku, Chiba 263-8522, Japan
- MR Author ID: 706593
- Email: fujikawa@math.s.chiba-u.ac.jp
- Katsuhiko Matsuzaki
- Affiliation: Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo 169-8050, Japan
- MR Author ID: 294335
- ORCID: 0000-0003-0025-5372
- Email: matsuzak@waseda.jp
- Received by editor(s): May 22, 2011
- Received by editor(s) in revised form: November 24, 2011
- Published electronically: January 4, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3309-3327
- MSC (2010): Primary 30F60, 32G15; Secondary 37F30, 30F25
- DOI: https://doi.org/10.1090/S0002-9947-2013-05767-4
- MathSciNet review: 3034467
Dedicated: Dedicated to Professor Masahiko Taniguchi on the occasion of his 60th birthday