Journal of the Mathematical Society of Japan

A uniqueness of periodic maps on surfaces

Susumu HIROSE and Yasushi KASAHARA

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Kulkarni showed that, if $g$ is greater than $3$, a periodic map on an oriented surface $\Sigma_g$ of genus $g$ with order not smaller than $4g$ is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if $g$ is greater than $30$, the same phenomenon happens for periodic maps on the surfaces with orders more than $8g/3$, and, for any integer $N$, there is $g > N$ such that there are periodic maps of $\Sigma_g$ of order $8g/3$ which are not conjugate up to power each other. Moreover, as a byproduct of our argument, we provide a short proof of Wiman's classical theorem: the maximal order of periodic maps of $\Sigma_g$ is $4g+2$.

Article information

J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1777-1787.

First available in Project Euclid: 24 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N05: Topology of $E^2$ , 2-manifolds
Secondary: 57M60: Group actions in low dimensions 20F38: Other groups related to topology or analysis

Riemann surface automorphism cyclic group


HIROSE, Susumu; KASAHARA, Yasushi. A uniqueness of periodic maps on surfaces. J. Math. Soc. Japan 68 (2016), no. 4, 1777--1787. doi:10.2969/jmsj/06841777.

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