On the number of \(p\)-gonal coverings of Riemann surfaces
A compact Riemann surface \(X\) of genus \(g\geq 2\) is said to be cyclic \(p\)-gonal if there exists an automorphism \(\phi\) of order \(p\) such that the orbit space \(X/\langle \phi \rangle\) has genus \(0\). Such an automorphism is called a \(p\)-gonal automorphism. If \(A\) is the full automorphism group of \(X\), then we call the group generated by \(\phi\) a cyclic \(p\)-gonal subgroup of \(A\). In the paper under review, the author determines a bound for the number of distinct cyclic \(p\)-gonal subgroups of \(A\). The author uses Fuchsian groups and uniformization together with Macbeath's theorem for the number of fixed points of an automorphism to derive the bounds, \textit{A. M. Macbeath} [Bull. Lond. Math. Soc. 5, 103--108 (1973; Zbl 0259.30016)]. The author is able to show that for \(p=3\) there are at most \(14\), for \(p=5\), at most \(12\) and for \(p\geq 7\), there are at most \(6(p-2)/(p-6)\) (so in general, there are at most \(30\)). Further consequences of these bounds are also discussed. Related literature studying the general structure of \(A\) can be found in [\textit{A. Kontogeorgis} [J. Algebra 216, No. 2, 665--706 (1999; Zbl 0938.11056)] where the isomorphism types of \(A\) are completely determined when the cyclic \(p\)-gonal subgroup is normal (in fact for general \(n\)), and \textit{A. Wootton} [J. Algebra 312, No. 1, 377--396 (2007; Zbl 1117.30034)] when the cyclic \(p\)-gonal subgroup is not normal.
scientific article; zbMATH DE number 5779825