Orientation reversing automorphisms of Riemann surfaces
The theme of the paper is up to what extent does, the set of fixed points of an automorphism on a compact Riemann surface \(X\) determine its conjugacy class. \textit{J. Nielsen} proved that this is so if we are dealing with orientation preserving automorphisms of prime order [Die Struktur periodischer Transformationen von Flächen, Math.-Fys. Medd., Danske Vid Selsk. 15, No. 1, 1-77 (1937; Zbl 0017.13302)]. In 1979 \textit{R. Zarrow} classified up to topological conjugation the orientation reversing automorphisms of order \(2p\), \(p\) prime, but the statements contain some mistakes. The result appears in two papers [``Orientation reversing square roots of involutions'' and ``Orientation reversing maps of surfaces'', both published in Ill. J. Math. 23, 71-81 and 82-92 (1979; Zbl 0407.30032 and Zbl 0407.30033)]. The goal of the paper under review is to obtain correctly such a classification. First it is shown, and this agrees with Zarrow's result, that the number of fixed points of the square, when different from zero, determines, up to conjugation, the orientation reversing automorphisms of order 4. Secondly, it is proved that if the genus \(g_ 0\) of \(X\) satisfies \(g_ 0\equiv 1\pmod 4\) there are two, instead of one as Zarrow claimed, conjugacy classes of orientation-reversing automorphisms of order 4 having squares without fixed points, and there are no such automorphisms if \(g_ 0\not\equiv 1\pmod 4\). The last result of the present paper states that the orientation- reversing automorphisms \(\varphi\) and \(\psi\) of order \(2p\), \(p\) an odd prime, are conjugate if and only if the following conditions hold: i) \(X/\varphi\) and \(X/\psi\) are homeomorphic, ii) \(\varphi^ 2\) and \(\psi^ 2\) are conjugate, iii) the action of \(\varphi^ 2\) on the set of fixed points of \(\varphi^ p\) is conjugate to the action of \(\psi^ 2\) on the set of fixed points of \(\psi^ p\). In Zarrow's paper this last condition is replaced by ``\(\varphi^ p\) and \(\psi^ p\) are conjugate''. However, the authors show the existence of two non-conjugate orientation-reversing automorphisms of order 10 satisfying i), ii) and Zarrow's condition. The proofs use some fundamental facts in the theory of compact Riemann surfaces and non-Euclidean crystallographic groups and some results concerning orientability [\textit{A. H. Hoare} and \textit{D. Singerman}, ``The orientability of subgroups of plane groups'' in Groups, Conf. St. Andrews 1981, Lond. Math. Soc. Lect. Notes 71, 221-227 (1982; Zbl 0489.20036)].
non-Euclidean crystallographic surface automorphism
scientific article; zbMATH DE number 622561