A uniqueness of periodic maps on surfaces (Q502105)

Denote by \(\Sigma_g\) an oriented closed surface of genus \(g\geq 2\). Let \(g > 30\), and \(n > (8/3)g\). It is proved (Theorem~1.1) that if there exists a periodic orientation preserving homeomorphism of \(\Sigma_g\) of order \(n\), then this map is unique up to conjugacy and power. The constant \((8/3)\) is sharp in the sense that for every \(N\geq 1\) there exists \(g > N\) such that there are periodic maps of \(\sigma_g\) of order \((8/3)g\) which are not conjugate, up to a power, to each other. Similar results with larger constants were obtained earlier by \textit{R. S. Kulkarni} [Contemp. Math. 201, 63--79 (1997; Zbl 0863.30050)] and \textit{S. Hirose} [Tohoku Math. J. (2) 62, No. 1, 45--53 (2010; Zbl 1198.57011)]. The authors give a table of genera and total valencies of homeomorphisms such that for any numbers \(g\) and \(n\) satisfying \(g>30\) and \(n > (8/3)g\), every periodic orientation preserving homeomorphism \(f\) of \(\Sigma_g\) having order \(n\) is conjugate to a power of a periodic map with characteristics listed in the table (Corollary 4.2).