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]]>

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 g _{1}< g_{2}*

*< ... 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**

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[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.

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

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

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