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Geometrically finite and semi-rational branched coverings of the two-sphere


Authors: Guizhen Cui and Yunping Jiang
Journal: Trans. Amer. Math. Soc. 363 (2011), 2701-2714
MSC (2010): Primary 37F20, 37F10, 30D05
DOI: https://doi.org/10.1090/S0002-9947-2010-05211-0
Published electronically: December 10, 2010
MathSciNet review: 2763733
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Abstract: In 1982, Thurston gave a necessary and sufficient condition for a critically finite branched covering of the two-sphere to itself to be combinatorially equivalent to a rational map. We discuss extending this result to geometrically finite rational maps. We give an example to show that Thurston's original condition is not sufficient. This example is topologically pathological near accumulation points of the postcritical set. We give two conditions forbidding such pathology, show that they are equivalent, and (in a sequel to the present paper) will show that Thurston's condition together with this tameness is both necessary and sufficient to characterize geometrically finite rational maps.


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Additional Information

Guizhen Cui
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: gzcui@math.ac.cn

Yunping Jiang
Affiliation: Department of Mathematics, Queens College of CUNY, Flushing, New York 11367 – and – Department of Mathematics, CUNY Graduate Center, New York, New York 10016
Email: Yunping.Jiang@qc.cuny.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05211-0
Keywords: Geometrically finite branched covering, semi-rational branched covering, sub-hyperbolic semi-rational branched covering
Received by editor(s): June 2, 2009
Received by editor(s) in revised form: September 18, 2009
Published electronically: December 10, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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