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Conformal Geometry and Dynamics

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The quasiconformal equivalence of Riemann surfaces and the universal Schottky space


Author: Hiroshige Shiga
Journal: Conform. Geom. Dyn. 23 (2019), 189-204
MSC (2010): Primary 30F60; Secondary 30C62, 30F40
DOI: https://doi.org/10.1090/ecgd/343
Published electronically: October 30, 2019
MathSciNet review: 4024932
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Abstract: In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite-type, the situation is rather complicated.

In this paper, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent.

Our argument enables us to obtain a universal property of the deformation spaces of Schottky regions, which is analogous to the fact that the universal Teichmüller space contains all Teichmüller spaces.


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Additional Information

Hiroshige Shiga
Affiliation: Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro-ku Tokyo, Japan
Address at time of publication: Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, Japan
Email: shiga@cc.kyoto-su.ac.jp

DOI: https://doi.org/10.1090/ecgd/343
Keywords: Riemann surface, quasiconformal map, Teichm\"uller spaces.
Received by editor(s): June 28, 2019
Received by editor(s) in revised form: July 25, 2019
Published electronically: October 30, 2019
Additional Notes: The author was partially supported by the Ministry of Education, Science, Sports and Culture, Japan; Grant-in-Aid for Scientific Research (B), 16H03933
Article copyright: © Copyright 2019 American Mathematical Society