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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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.


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Additional Information

Ege Fujikawa
Affiliation: Department of Mathematics, Chiba University, Inage-ku, Chiba 263-8522, Japan
Email: fujikawa@math.s.chiba-u.ac.jp

Katsuhiko Matsuzaki
Affiliation: Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo 169-8050, Japan
Email: matsuzak@waseda.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05767-4
PII: S 0002-9947(2013)05767-4
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.