SciELO - Scientific Electronic Library Online

 
vol.20 número1ON THE LEVI PROBLEM WITH SINGULARITIES índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

Compartir


Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.20 n.1 Antofagasta mayo 2001

http://dx.doi.org/10.4067/S0716-09172001000100007 

SCHOTIKY UNIFORMIZATIONS
AND RIEMANN MATRICES OF
MAXIMAL SYMMETRIC RIEMANN
SURFACES OF GENUS 5*

 

RUBÉN HIDALGO
Universidad Técnica Federico Santa María-Chile    

Abstract

In this note we consider pairs (S, T), where S is a closed Riemann surface of genus five and T : S ®   S in some anticonformal involution with fixed points so that K(S, T) = {h Î  Aut±(S) :  hT =  Th} has the maximal order 96 and S /TT  is orientable.  We observe that there are exactly two topologically different choices for T. They give non-isomorphic groups K(S,T), each one acting topologically rigid on the respective surface S. These two cases give then two (connect) real algebraic sets of real dimension one in the moduli space of genus 5. In this note we describe these components by classical Schottky groups and with the help of these uniformizations we compute their Riemann matrices.

Keywords: Schottky groups, Riemann surfaces, Riemann matrices.

Subjclass: Primary 30F40.
_____________________________
*Partially supported by projects UTFSM 12.01.22, Fondecyt 1000715 and a Presidential Chair on Geometry.  

 

References

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

[2] Burnside, W.: On a class of  Automorphic Functions. Proc. London Math. Soc. Vol 23 (1892), 49-88.         [ Links ]

[3] Conway, J.H. and Sloan, N. J. A.: it Sphere Packings, Lattices and Groups. Springer-Verlag, 1988.         [ Links ]

[4] Costa, A.F. and Hidalgo, R.A.: Anticonformal automorphisms and Schottky coverings.  Preprint.         [ Links ]

[5] Chuckrow, V.: On Schottky groups with application to Kleinian groups. Ann. of Math. 88 (1968), 47-61.

[6] González-Díz, G.: Loci of curves which are prime Galois coverings of P1. Proc. London Math. Soc. 62 (1991), 469-489.

[7] Heltai, B.: Symmetric Riemann surfaces, torsion subgroups and Schottky coverings. Proc. Amer. Math. Soc. 100 (1987), 675-682.

[8] Hidalgo, R.A.: On Schottky groups with automorphisms. Ann. Acad. Scie. Fenn. Ser. AI Mathematica 19 (1994), 259-289.         [ Links ]

[9]  Hidalgo, R.A.: Schottky uniformizations of closed Riemann surfaces with Abelian  groups of conformal automorphisms. Glasgow Math. J. 36 (1994), 17-32.         [ Links ]

[10] Hidalgo, R.A.: Dihedral groups are of Schottky type.  Revista Proyecciones. 18 (1999), 23-48.         [ Links ]

[11] Hidalgo, R.A.: A4, A5 and S4 of Schottky type. Preprint.

[12] Hidalgo, R.A.: Bounds for Conformal Automorphisms of Riemann Surfaces with Condition (A). Preprint.         [ Links ]

[13] Keen, L.: On hyperelliptic Schottky groups. Ann. Acad. Sci. Fenn. Series A.I. Mathematica 5 (1980).         [ Links ]

[14] Marden, A.: Schottky groups and circles.  Contribution to Analysis, A collection of papers Dedicated to Lipman Bers. (1994), 273-278.         [ Links ]

[15] Maskit, B.: Kleinian Groups. G.M.W. 287, Springer-Verlag, 1988.

[16] Maskit, B.: Special uniformizations of symmetric Riemann surfaces. Preprint.         [ Links ]

[17] Maskit, B.: A characterization of Schottky groups. J. d'Analyse Math. 19 (1967), 227-230.         [ Links ]

[18] May, C.L.: Automorphisms of compact Klein surfaces with boundary. Pacific J. Math. 59 (1975), 199-210.         [ Links ]

[19] May, C.L.: A bound for the number of automorphisms of a compact Klein surface with boundary. Proc. Amer. Math. Soc. 63 (1977), 273-280.

[20] Mumford, D.: Tata lectures on Theta II. Progress on Mathematics 43, Birkhaüser, Boston, 1984.         [ Links ]

[21] Natanzon, S.M.: Differential equations for Prym theta-functions, a criterion for two-dimensional finite zone potential Schrödinger operators to be real.  Funktsional Anal. i Prilozhen 26 (1992), 17-26.         [ Links ]

[22] Quine, J.R. and Zhang, P.: Extremal symplectic lattices. Preprint.         [ Links ]

[23] Schmutz, P.: Riemann surfaces with shortest geodesic of maximal lenght, Gemetric and Functional Analysis, 3 (1993), 564-631.         [ Links ]

[24] Sibner, R.J.: Uniformization of Symmetric Riemann surfaces by Schottky groups. Trans. Amer. Math. Soc. 116 (1965), 79-85.         [ Links ]

Received: November 2000.

Rubén A. Hidalgo
Departamento de Matemática
Universidad Técnica Federico Santa María
Valparaíso
Chile
e-mail: rhidalgo@mat.utfsm.cl

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons