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

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## Abstract:

For an analytically infinite Riemann surface $R$, the quasiconformal mapping class group $\operatorname {MCG}(R)$ always acts faithfully on the ordinary Teichmüller space $T(R)$. However in this paper, an example of $R$ is constructed for which $\operatorname {MCG}(R)$ acts trivially on its asymptotic Teichmüller space $AT(R)$.## 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. Earle, F. Gardiner and N. Lakic,
*Teichmüller spaces with asymptotic conformal equivalence*, I.H.E.S. Preprint (1995). - 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 - Ege Fujikawa,
*The action of geometric automorphisms of asymptotic Teichmüller spaces*, Michigan Math. J.**54**(2006), no. 2, 269–282. MR**2252759**, DOI 10.1307/mmj/1156345593 - 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 - 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 - 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 - K. Matsuzaki,
*Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces*, J. Analyse Math., to appear.

## 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
- MR Author ID: 294335
- ORCID: 0000-0003-0025-5372
- Email: matsuzak@math.okayama-u.ac.jp
- 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
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2302578