Abstract
Let X be a compact Riemann surface of genus g ≥ 2. A cyclic subgroup of prime order p of Aut(X) is called properly (p, h)-gonal if it has a fixed point and the quotient surface has genus h. We show that if p > 6h + 6, then a properly (p, h)-gonal subgroup of Aut(X) is unique. We also discuss some related results.
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R. Accola, Topics in the Theory of Riemann Surfaces, Springer Lecture Notes in Mathematics 1595, Berlin-Heidelberg-New York, 1994
Belolipetsky M., Jones G.A.: Automorphism groups of Riemann surfaces of genus p + 1, where p is prime, Glasgow Math. J. 47, 379–393 (2005)
T. Breuer, Characters and Automorphism Groups of Compact Riemann Surfaces, LMS Lecture Notes 280, Cambridge University Press, Cambridge, 2000
Bujalance E., Conder M.: On cyclic groups of automorphisms of Riemann surfaces. J. London Math. Soc. 59, 573–584 (1999)
Costa A.F., Izquierdo M., Ying D.: On Riemann surfaces with non-unique cyclic trigonal morphism, Manuscripta Math. 118, 443–453 (2005)
A. F. Costa and H. Parlier, Applications of a theorem of Singerman about Fuchsian groups, Arch. Math. (Basel) 91 (2008), 536–543
Farkas H. M., Kra I.: Riemann Surfaces. Springer, Berlin-Heidelberg-New York (1980)
González-Diez G.: On Prime Galois Coverings of the Riemann Sphere. Ann. Mat. Pura Appl. 168, 1–15 (1995)
Gromadzki G.: On Conjugacy of p-gonal Automorphisms of Riemann Surfaces. Rev. Mat. Complut. 21, 83–87 (2008)
Gromadzki G.: On the number of p-gonal coverings of Riemann surfaces. Rocky Mountain J. Math. 40, 1221–1226 (2010)
G. Gromadzki, R Hidalgo., On prime Galois coverings of tori, preprint.
Gromadzki G., Weaver A., Wootton A.: On gonality of Riemann surfaces. Geom. Dedicata 194, 1–14 (2012)
Hidalgo R.: On conjugacy of p-gonal automorphisms, Bull. Korean Math. Soc. 49, 411–415 (2012)
Wootton A.: The full automorphism group of a cyclic p-gonal surface. J. Algebra 312, 377–396 (2007)
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This paper was written while the author was supported by grant 99-2115-M-001-011-MY2 from the National Science Council (NSC) of Taiwan.
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Schweizer, A. On the uniqueness of (p, h)-gonal automorphisms of Riemann surfaces. Arch. Math. 98, 591–598 (2012). https://doi.org/10.1007/s00013-012-0397-8
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DOI: https://doi.org/10.1007/s00013-012-0397-8