A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space

Author:
Katsuhiko Matsuzaki

Journal:
Proc. Amer. Math. Soc. **135** (2007), 2573-2579

MSC (2000):
Primary 30F60; Secondary 32G15

DOI:
https://doi.org/10.1090/S0002-9939-07-08761-8

Published electronically:
March 22, 2007

MathSciNet review:
2302578

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Abstract: For an analytically infinite Riemann surface , the quasiconformal mapping class group always acts faithfully on the ordinary Teichmüller space . However in this paper, an example of is constructed for which acts trivially on its asymptotic Teichmüller space .

**[1]**C. Earle and F. Gardiner,*Geometric isomorphisms between infinte dimensional Teichmüller spaces*, Trans. Amer. Math. Soc.**348**(1996), 1163-1190. MR**1322950 (96h:32024)****[2]**C. Earle, F. Gardiner and N. Lakic,*Teichmüller spaces with asymptotic conformal equivalence*, I.H.E.S. Preprint (1995).**[3]**C. Earle, F. Gardiner and N. Lakic,*Asymptotic Teichmüller space, Part I: the complex structure*, In the tradition of Ahlfors and Bers, Contemp. Math., vol. 256, AMS, 2000, pp. 17-38. MR**1759668 (2001m:32029)****[4]**C. Earle, F. Gardiner and N. Lakic,*Asymptotic Teichmüller space, Part II: the metric structure*, In the tradition of Ahlfors and Bers III, Contemp. Math., vol. 355, AMS, 2004, pp. 187-219. MR**2145063 (2006g:30078)****[5]**C. Earle, V. Markovic and D. Saric,*Barycentric extension and the Bers embedding for asymptotic Teichmüller space*, Complex manifolds and hyperbolic geometry, Contemp. Math., vol. 311, AMS, 2002, pp. 87-105. MR**1940165 (2003i:30072)****[6]**E. Fujikawa,*The action of geometric automorphisms of asymptotic Teichmüller spaces*, Michigan Math. J.**54**(2006), 269-282. MR**2252759****[7]**F. Gardiner and D. Sullivan,*Symmetric structures on a closed curve*, Amer. J. Math.**114**(1992), 683-736. MR**1175689 (95h:30020)****[8]**V. Markovic,*Biholomorphic maps between Teichmüller spaces*, Duke Math. J.**120**(2003), 405-431. MR**2019982 (2004h:30058)****[9]**K. Matsuzaki,*A countable Teichmüller modular group*, Trans. Amer. Math. Soc.**357**(2005), 3119-3131. MR**2135738 (2006f:30052)****[10]**K. Matsuzaki,*Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces*, J. Analyse Math., to appear.

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Additional Information

**Katsuhiko Matsuzaki**

Affiliation:
Department of Mathematics, Ochanomizu University, Tokyo 112-8610, Japan

Address at time of publication:
Department of Mathematics, Okayama University, Okayama 700-8530, Japan

Email:
matsuzak@math.okayama-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-07-08761-8

Keywords:
Analytically infinite Riemann surface,
quasiconformal mapping class group,
asymptotic Teichm\"{u}ller space

Received by editor(s):
August 16, 2005

Received by editor(s) in revised form:
April 19, 2006

Published electronically:
March 22, 2007

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.