On classification of cyclic orientation-reversing actions of big order on closed surfaces
Let \(S_g\) be a closed orientable surface of genus \(g \geq 2\). Since the 19\({}^{\mathrm{th}}\) century it is known that the order of an orientation-preserving self-homeomorphism of \(S_g\) is at most \(4g+2\). Concerning orientation-reversing homeomorphisms, the reviewer proved in [Arch. Math. 45, 374--384 (1985; Zbl 0561.20037)] that their order is at most \(4g+4\) for even \(g\), and \(4g-4\) for odd \(g\), and that both bounds are attained for every \(g\). In the paper under review, the authors consider all orientation-reversing actions of order \(N\) provided that \(N > 2g-2\). Their goal is to compute the number of conjugacy classes of such actions up to topological conjugation. The main tool in order to get it is the uniformization of Riemann and Klein surfaces by means of Fuchsian and NEC groups. Then, for \(N > 2g-2\), the authors split the study into the seven possible signatures for the involved NEC group, or equivalently, into the seven possible orbifolds \(S_g/\mathbb{Z}_N\). The complete result is displayed in Table 1, whilst Table 2 particularizes it for \(N \geq 3g-3\). In particular, this specification implies that, excluding \(g = 2, 4, 6, 10\), the three biggest possible orders, namely \(4g+4\), \(4g\) and \(4g-4\), determine a unique class up to topological conjugation.
scientific article; zbMATH DE number 6499150