Journal of the Mathematical Society of Japan

Degenerations and fibrations of Riemann surfaces associated with regular polyhedra and soccer ball


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To each of regular polyhedra and a soccer ball, we associate degenerating families (degenerations) of Riemann surfaces. More specifically: To each orientation-preserving automorphism of a regular polyhedron (and also of a soccer ball), we associate a degenerating family of Riemann surfaces whose topological monodromy is the automorphism. The complete classification of such degenerating families is given. Besides, we determine the Euler numbers of their total spaces. Furthermore, we affirmatively solve the compactification problem raised by Mutsuo Oka — we explicitly construct compact fibrations of Riemann surfaces that compactify the above degenerating families. Their singular fibers and Euler numbers are also determined.

Article information

J. Math. Soc. Japan, Volume 69, Number 3 (2017), 1213-1233.

First available in Project Euclid: 12 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D06: Fibrations, degenerations
Secondary: 57M99: None of the above, but in this section 58D19: Group actions and symmetry properties

degenerating family of Riemann surfaces fibration monodromy regular polyhedron group action automorphism group cyclic quotient singularity


HIRAKAWA, Ryota; TAKAMURA, Shigeru. Degenerations and fibrations of Riemann surfaces associated with regular polyhedra and soccer ball. J. Math. Soc. Japan 69 (2017), no. 3, 1213--1233. doi:10.2969/jmsj/06931213.

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