?-hyperelliptic Riemann surfaces


  • Rubén A. Hidalgo Universidad Técnica Federico Santa María.




Puchsian groups, Schottky groups, Riemann surfaces


We give some characterizations of ? -hyperelliptic Riemann surfaces of genus ? ? 2, that is, pairs (S, j) where S is a closed Riemann surface of genus ? and j : S ? S is a conformal involution with exactly 2? + 2 - 4?  fixed points. These characterizations are given by Schottky groups, special hyperbolic polygons and algebraic curves.

These can be seen as generalizations of the works [5] and [11].

Author Biography

Rubén A. Hidalgo, Universidad Técnica Federico Santa María.

Departamento de Matemática.


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

[2] Bers, L., Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14, pp. 215-228, (1961).

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

[4] Farkas, H. and Kra, I., "Riemann surfaces", Springer-Verlag.

[5] Gallo, D., Hyperelliptic Riemann surfaces, thesis Ph.D. S.U.N.Y. at Stony Brook, (1979).

[6] Haas, A. and Susskind,. P., The geometry of the hyperelliptic involution in genus two,

[7] Hejhal, D. A., On Schottky and Teichmüller spaces, Adv. in Math. 15, pp. 133-156, (1975).

[8] Hidalgo, R., On ? -hyperelliptic Schottky groups, Boletín de la Sociedad de Matemática de Chile, 8, pp. 27-36, (1989).

[9] Hidalgo, R., On Schottky Groups with Automorphisms, thesis Ph.D. in Math. S.U.N.Y. at Stony Brook, New York, (1991).

[10] Hiro-0 Yamamoto, An example of a nonclassical Schottky group, Duke Math. J. 63, pp. 193-197, (1991).

[11] Keen, L., On hyperelliptic Schottky groups, Annales Academiae Scientianun Fennicae. Series A.I. Mathematica, volumen 5, pp. 165-174, (1980).

[12] Keen, L., Canonical polygons for finitely generated Fuchsian groups, Acta Mathematica 115, (1966).

[13] Maskit, B., Kleinian groups, Springer-Verlag.

[14] Maskit, B., Decomposition of certain Kleinian groups, Acta Mathematica, vol 130, (1973).

[15] Marden, A., Schottky groups and circles, in Contribution to Analysis: A collection of Papers Dedicated to Lipman Bers, ed. Lars V. Ahlfors, et. al., Academic Press, New York, pp. 273-278, (1964).

[16] Whittaker, E., On the connection of algebraic functions with automorphic functions, Phil. Trans., vol 192, pp. 1-32, (1899)



How to Cite

R. A. Hidalgo, “?-hyperelliptic Riemann surfaces”, Proyecciones (Antofagasta, On line), vol. 17, no. 1, pp. 77-117, Apr. 2018.