Limit sets and regions of discontinuity of Teichmüller modular groups

Author:
Ege Fujikawa

Journal:
Proc. Amer. Math. Soc. **132** (2004), 117-126

MSC (2000):
Primary 30F60; Secondary 30C62

DOI:
https://doi.org/10.1090/S0002-9939-03-06988-0

Published electronically:
February 28, 2003

MathSciNet review:
2021254

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Abstract | References | Similar Articles | Additional Information

Abstract: For a Riemann surface of infinite type, the Teichmüller modular group does not act properly discontinuously on the Teichmüller space, in general. As an analogy to the theory of Kleinian groups, we divide the Teichmüller space into the limit set and the region of discontinuity for the Teichmüller modular group, and observe their properties.

**1.**A. F. Beardon,*The Geometry of Discrete Groups*, Graduate Texts in Mathematics**91**, Springer, 1983. MR**85d:22026****2.**C. J. Earle, F. P. Gardiner and N. Lakic,*Teichmüller spaces with asymptotic conformal equivalence,*preprint.**3.**A. Epstein,*Effectiveness of Teichmüller modular groups*, In the tradition of Ahlfors and Bers, Contemporary Math. 256, American Mathematical Society, 2000, 69-74. MR**2001a:30059****4.**E. Fujikawa, H. Shiga and M. Taniguchi,*On the action of the mapping class group for Riemann surfaces of infinite type*, J. Math. Soc. Japan, to appear.**5.**F. P. Gardiner,*Teichmüller theory and quadratic differentials*, Wiley-Interscience, New York, 1987. MR**88m:32044****6.**F. Hausdorff,*Set Theory*, Third Edition, Chelsea Publishing Company, New York, 1978.**7.**Y. Imayoshi and M. Taniguchi,*Introduction to Teichmüller Spaces*, Springer-Tokyo 1992. MR**94b:32031****8.**J. P. Matelski,*A compactness theorem for Fuchsian groups of the second kind*, Duke Math. J.**43**(1976), 829-840. MR**55:5900****9.**K. Matsuzaki and M. Taniguchi,*Hyperbolic Manifolds and Kleinian Groups*, Oxford Science Publications, 1998. MR**99g:30055****10.**S. Nag,*The Complex Analytic Theory of Teichmüller Spaces*, John Wiley & Sons, 1988. MR**89f:32040****11.**M. Tsuji,*Potential Theory in Modern Function Theory*, Chelsea, New York, 1959.**12.**S. A. Wolpert,*The length spectra as moduli for compact Riemann surfaces*, Ann. Math.**109**(1979), 323-351. MR**80j:58067**

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Additional Information

**Ege Fujikawa**

Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama Meguro-ku Tokyo 152-8551, Japan

Email:
fujikawa@math.titech.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-03-06988-0

Keywords:
Infinite dimensional Teichm\"uller space,
Teichm\"uller modular groups,
hyperbolic geometry

Received by editor(s):
August 12, 2002

Published electronically:
February 28, 2003

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2003
American Mathematical Society