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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.20 n.2 Antofagasta ago. 2001

http://dx.doi.org/10.4067/S0716-09172001000200002 

BOUNDS FOR CONFORMAL
AUTOMOMORPHISMS OF RIEMANN
SURFACES WITH CONDITION (A) *

RUBÉN A. HIDALGO
Universidad Técnica Federico Santa María, Chile

 

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 ³ 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.

Subjclass : [2000] Primary 30F10, 30F40
Keywords : Schottky groups, Reimann surfaces, conformal automorphisms



*Partially supported by projects UTFSM 12.01.22, Fondecyt 1000715 and Fondecyt 1010093.


REFERENCES

 

[1] L. V. Ahlfors and L. Sario. Riemann surfaces. Princeton Univ. Press. Princeton, New Jersey, (1960).        [ Links ]

[2] L. Bers. Automorphic forms for Schottky groups. Adv. in Math., 16, pp. 332-361, (1975).        [ Links ]

[3] V. Chuckrow. On Schottky groups with applications to Kleinian groups. Ann. of Math. 88, pp. 47-61, (1968).        [ Links ]

[4] H. Farkas and I. Kra. Riemann surfaces. Springer - Verlag, New York. (1980).        [ Links ]

[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).        [ Links ]

[6] V. González and R. Rodríguez. On automorphisms of compact Riemann surfaces of genus four. Complex geometry seminar, volume II, UTFSM, (1986).        [ Links ]

[7] R. Hidalgo. Schottky Uniformizations of closed Riemann surfaces with Abelian groups of conformal automorphisms. Glasgow Math. J. bf 35, pp. 17-32, (1993)..         [ Links ]

[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.).        [ Links ]

[9] R. Hidalgo. Dihedral groups are of Schottky type. Preprint.         [ Links ]

[10] R. Hidalgo A4, A5 and S4 of Schottky type. Preprint.        [ Links ]

[11] S. Kerckhoff. The Nielsen realization problem. Annals of Mathematics, 177, pp. 235-265, (1983).        [ Links ]

[12] A.M. Macbeath. On a theorem of Hurwitz. Proc. Glasgow Math. Assoc., 5, pp. 90-96, (1961).        [ Links ]

[13] B. Maskit. Kleinian groups. Grundlehren der Mathematischen Wissenschaften, volume 287, Springer - Verlag, Berlin, Heildelberg, New York, (1988).        [ Links ]

[14] B. Maskit. A characterization of Schottky groups. J. d´Analyse Math., 19, pp. 227-230, (1967).        [ Links ]

[15] D. McCullough, A. Miller and B. Zimmermann. Group actions on Handlebodies. Proc. London Math. Soc., 59, pp. 373-416, (1989).        [ Links ]

[16] A. Miller and B. Zimmermann. Large groups of symmetries of handlebodies. Proc. Amer. Math. Soc., 106, pp.829-838, (1989).        [ Links ]

[17] K. Nakagawa. On the orders of automorphisms of a closed Riemann surface. Pacific J. of Math., 115, (1984).        [ Links ]

[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).        [ Links ]

[19] B. Zimmermann. Über Homöomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen. Comment. Math. Helv., 56, pp. 474-486, (1981).        [ Links ]

Received : May, 2000.

RUBEN A. HIDALGO
Departamento de Matemática
Universidad Técnica Federico Santa María
Casilla 110-V
Valparaíso
Chile
E-MAIL: rhidalgo@mat.utfsm.cl