Riemann surfaces with real forms which have maximal cyclic symmetry
It was proved by \textit{C. L. May} [Cyclic automorphism groups of compact bordered Klein surfaces, Houston J. Math. 3, 395--405 (1977; Zbl 0379.14012)] and \textit{E. Bujalance} [Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Pac. J. Math. 109, 279--289 (1983; Zbl 0545.30033)] that the maximum order of an automorphism of a compact Klein surface of algebraic genus \(g\geq 2\) is \(\mu(g)= 2g+1+(-1)^g\). The authors of the paper under review prove that, unlike the classical case, there exist non-isomorphic Klein surfaces for which this upper bound is attained. More precisely, they describe explicitely the moduli space of compact Riemann surfaces \(X\) having a symmetry \(\tau\) such that the compact Klein surface \(Y= X/\tau\) admits an automorphism of maximum order \(\mu(g)\) as a real one dimensional semianalytic space. Another striking results proves that, with the notations above, the full automorphism group of \(Y\) is not cyclic; even more, with a finite number of exceptions, it is dihedral and coincides with the full automorphism group of its double cover. This work, whose proofs and structure are limpid, provides also algebraic equations of all involved objects, i.e, the genus \(g\) Riemann surface \(X\), the symmetry \(\tau\), the cyclic automorphism of maximum order u\(\mu(g)\) of \(Y= X/\tau\), and a set of generators of the full automorphism group of \(Y\). To finish, let us point out that the article contains also some interesting contributions to the knowledge of the funcion \(v:\{g\in \mathbb{Z},g\geq 2\}\to\mathbb{N}\), where \(v(g)\) is the order of the largest group of automorphisms of a compact Klein surface of algebraic genus \(g\).
scientific article; zbMATH DE number 2136591
scientific article