Bounds for conformal automorphisms of riemann surfaces with condition (A)
DOI:
https://doi.org/10.4067/S0716-09172001000200002Keywords:
Schottky groups, Riemann surfaces, Conformal automorphisms.Abstract
In this note we consider a class of groups of conformal automorphisms of closed Riemann surfaces containing those which can be lifted to some Schottky uniformization. These groups are those which satisfy a necessary condition for the Schottky lifting property. We find that all these groups have upper bound 12(g ? 1), where g ? 2 is the genus of the surface. We also describe a sequence of infinite genera g1 < g2 < · · · for which these upper bound is attained. Also lower bounds are found, for instance, (i) 4(g+1) for even genus and 8(g?1) for odd genus. Also, for cyclic groups in such a family sharp upper bounds are given.References
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[2] L. Bers. Automorphic forms for Schottky groups. Adv. in Math., 16, pp. 332–361, (1975).
[3] V. Chuckrow. On Schottky groups with applications to Kleinian groups. Ann. of Math. 88, pp. 47–61, (1968).
[4] H. Farkas and I. Kra. Riemann surfaces. Springer - Verlag, New York, (1980).
[5] L. Keen, B. Maskit and C. Series. Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets. J. reine angew. Math. 436, pp. 209–219, (1993).
[6] V. González and R. Rodríguez. On automorphisms of compact Riemann surfaces of genus four. Complex geometry seminar, volume II, UTFSM, (1986).
[7] R. Hidalgo. Schottky Uniformizations of closed Riemann surfaces with Abelian groups of conformal automorphisms. Glasgow Math. J. bf 35, pp. 17–32, (1993).
[8] R. Hidalgo. On Schottky groups with automorphisms. Theses Ph. D. of Mathematics, S.U.N.Y. at Stony Brook, (1991). (to be published on Ann. Acad. Sci. Fenn.).
[9] R. Hidalgo. Dihedral groups are of Schottky type. Preprint.
[10] R. Hidalgo. A4, A5 and S4 of Schottky type. Preprint.
[11] S. Kerckhoff. The Nielsen realization problem. Annals of Mathematics, 117, pp. 235-265, (1983).
[12] A.M. Macbeath. On a theorem of Hurwitz. Proc. Glasgow Math. Assoc., 5, pp. 90–96, (1961).
[13] B. Maskit. Kleinian groups. Grundlehren der Mathematischen Wissenschaften, volume 287, Springer - Verlag, Berlin, Heildelberg, New York, (1988).
[14] B. Maskit. A characterization of Schottky groups. J. d’Analyse Math., 19, pp. 227–230, (1967).
[15] D. McCullough, A. Miller and B. Zimmermann. Group actions on Handlebodies. Proc. London Math. Soc., 59, pp. 373-416, (1989).
[16] A. Miller and B. Zimmermann. Large groups of symmetries of handlebodies. Proc. Amer. Math. Soc., 106, pp. 829–838, (1989).
[17] K. Nakagawa. On the orders of automorphisms of a closed Riemann surface. Pacific J. of Math., 115, (1984).
[18] R. Ruedy. Symmetric embeddings of Riemann surfaces. In Discontinuous groups and Riemann surfaces. Ed., by Leon Greenberg. Annals of Math. Studies. Princeton Univ. Press., number 79, (1974).
[19] B. Zimmermann. Uber Homömorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen. Comment. Math. Helv., 56, pp. 474-486, (1981).
Published
2017-04-24
How to Cite
[1]
R. A. Hidalgo, “Bounds for conformal automorphisms of riemann surfaces with condition (A)”, Proyecciones (Antofagasta, On line), vol. 20, no. 2, pp. 139-175, Apr. 2017.
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