Wootton, Aaron Multiple prime covers of the Riemann sphere. (English) Zbl 1108.30030 Cent. Eur. J. Math. 3, No. 2, 260-272 (2005). A cyclic \(q\)-gonal surface is a compact Riemann surface \(X\) of genus \(\geq 2\) which admits a cyclic group of automorphisms \(C_q\) of prime order \(q\) such that \(X/C_q\) has genus \(0\). The group \(C_q\) is called the \(q\)-gonal group for \(X\). If a \(q\)-gonal surface \(X\) is also a \(p\)-gonal surface for some prime \(p\not=q\), then \(X\) is called a multiple prime surface. The aim of this paper is to classify all multiple prime surfaces, that is, to find the automorphism group and the signature for the normalizer of a surface group for each such surface. As an application, the author proves the following:Theorem: A cyclic \(q\)-gonal surface can be \(p\)-gonal for at most one other prime \(p\).Theorem: If \(X\) is a multiple prime surface which is cyclic \(q\)-gonal and cyclic \(p\)-gonal, then any element from a \(q\)-gonal group commutes with any element from a \(p\)-gonal group. Reviewer: Athanase Papadopoulos (Strasbourg) Cited in 6 Documents MSC: 30F10 Compact Riemann surfaces and uniformization 14H37 Automorphisms of curves 14H30 Coverings of curves, fundamental group 30F60 Teichmüller theory for Riemann surfaces Keywords:Riemann surface; automorphism; prime cover PDFBibTeX XMLCite \textit{A. Wootton}, Cent. Eur. J. Math. 3, No. 2, 260--272 (2005; Zbl 1108.30030) Full Text: DOI References: [1] R.D.M. Accola: “Strongly Branched Covers of Closed Riemann Surfaces”, Proc. of the AMS, Vol. 26(2), (1970), pp. 315-322. http://dx.doi.org/10.2307/2036396; · Zbl 0212.42501 [2] R.D.M. Accola: “Riemann Surfaces with Automorphism Groups Admitting Partitions”, Proc. Amer. Math. Soc., Vol. 21, (1969), pp. 477-482. http://dx.doi.org/10.2307/2037029; · Zbl 0174.37401 [3] T. Breuer: Characters and Automorphism Groups of Compact Riemann Surfaces, Cambridge University Press, 2001.; · Zbl 0952.30001 [4] E. Bujalance, F.J. Cirre and M.D.E. Conder: “On Extendability of Group Actions on Compact Riemann Surfaces”, Trans. Amer. Math. Soc., Vol. 355, (2003), pp. 1537-1557. http://dx.doi.org/10.1090/S0002-9947-02-03184-7; · Zbl 1019.20018 [5] A.M. Macbeath: “On a Theorem of Hurwitz”, Proceedings of the Glasgow Mathematical Association, Vol. 5, (1961), pp. 90-96. http://dx.doi.org/10.1017/S2040618500034365; · Zbl 0134.16603 [6] B. Maskit: “On Poincaré”s Theorem for Fundamental Polygons“, Advances in Mathematics, (1971), Vol. 7, pp. 219-230. http://dx.doi.org/10.1016/S0001-8708(71)80003-8<pub-id pub-id-type=”doi”>10.1016/S0001-8708(71)80003-8; · Zbl 0223.30008 [7] D. Singerman: “Finitely Maximal Fuchsian Groups”, J. London Math. Soc., Vol. 2 (6), (1972), pp. 29-38.; · Zbl 0251.20052 [8] A. Wootton: “Non-Normal Belyî p-gonal Surfaces”, In: Computational Aspects of Algebraic Curves, Lect. Notes in Comp., (2005), to appear.; · Zbl 1200.14054 [9] A. Wootton: “Defining Algebraic Polynomials for Cyclic Prime Covers of the Riemann Sphere”, Dissertation, (2004).; · Zbl 1109.30036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.