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Conformal Geometry and Dynamics

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Cusps in complex boundaries of one-dimensional Teichmüller space

Author: Hideki Miyachi
Journal: Conform. Geom. Dyn. 7 (2003), 103-151
MSC (2000): Primary 30F40, 30F60; Secondary 37F30, 37F45
Published electronically: September 9, 2003
MathSciNet review: 2023050
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Abstract: This paper gives a concrete proof of the conjectural phenomena on the complex boundary one-dimensional slices: every rational boundary point is cusp shaped. This paper treats this problem for Bers slices, the Earle slice, and the Maskit slice. In proving this, we also show that every Teichmüller modular transformation acting on a Bers slice can be extended as a quasi-conformal mapping on its ambient space. Furthermore, using this extension, we discuss similarity phenomena on the boundaries of Bers slices, and also compare these phenomena with results in complex dynamics. We will also give a result, related to the theory of L. Keen and C. Series, of pleated varieties in quasifuchsian space of once punctured tori.

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Additional Information

Hideki Miyachi
Affiliation: Department of Mathematics, Osaka City University, Sumiyoshi, Osaka, 558-8585, Japan
Address at time of publication: Graduate School of Science, Department of Mathematics, Osaka University, 1-1, Machikaneyama-cho, Toyonaka, Osaka, 560-0023, Japan

Keywords: Kleinian group, Teichm\"{u}ller theory, deformation space
Received by editor(s): July 2, 2000
Received by editor(s) in revised form: February 20, 2003
Published electronically: September 9, 2003
Additional Notes: This work was done when the author was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Dedicated: Dedicated to my father Kenji Miyachi for his 60th Birthday
Article copyright: © Copyright 2003 American Mathematical Society

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