On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

Ewa Tyszkowska

Open Mathematics, Tome 1 (2003), p. 208-220 / Harvested from The Polish Digital Mathematics Library

Résumé

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p n are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most

Détail

Publié le : 2003-01-01

Zbl 1074.30040

EUDML-ID : urn:eudml:doc:268693

Bibliographie

[1] G.V. Belyi: “On Galois extensions of maximal cyclotomic field” (English translation), Math. USSR Izvestiya, Vol. 14, (1980), pp. 247–256. http://dx.doi.org/10.1070/IM1980v014n02ABEH001096 | Zbl 0429.12004

[2] E. Bujalance, J. Etayo, J. Gamboa, G. Gromadzki: “Automorphisms Groups of Compact Bordered Klein Surfaces. A Combinatorial Approach”, Lecture Notes in Math., Vol. 1439, Springer Verlag, 1990. | Zbl 0709.14021

[3] S.A. Broughton, E. Bujalance, A.F. Costa, J.M. Gamboa, G. Gromadzki: “Symmetries of Riemann surfaces on which PSL(2,q) acts as a Hurwitz automorphism group”, J. Pure Appl. Alg., Vol. 106, (1996), pp. 113–126. http://dx.doi.org/10.1016/0022-4049(94)00065-4 | Zbl 0847.30026

[4] P. Buser and M. Seppälä: “Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves”, Math. Z., Vol. 232, (1999), pp. 547–558. http://dx.doi.org/10.1007/PL00004771 | Zbl 0948.32016

[5] H. Glover and D. Sjerve: “Representing PSL(2,p) on a Riemann surface of least genus”, L'Enseignement Mathematique, Vol. 31, (1985), pp. 305–325. | Zbl 0594.30046

[6] H. Glover and D. Sjerve: “The genus of PSL(2,q)”, Jurnal fur die Reine und Angewandte Mathematik, Vol. 380, (1987), pp. 59–86. http://dx.doi.org/10.1515/crll.1987.380.59 | Zbl 0613.57019

[7] G. Gromadzki: “On a Harnack-Natanzon theorem for the family of real forms of Rieamnn surfaces”, J. Pure Appl. Alg., Vol. 121, (1997), pp. 253–269. http://dx.doi.org/10.1016/S0022-4049(96)00068-0 | Zbl 0885.14026

[8] A. Harnack: “Uber die Vieltheiligkeit der ebenen algebraischen Kurven”, Math. Ann., Vol. 10, (1876), pp. 189–199. http://dx.doi.org/10.1007/BF01442458

[9] U. Langer and G. Rosenberger: “Erzeugende endlicher projectiver linearer Gruppen”, Result in Mathematics, Vol. 15, (1989), pp. 119–148. | Zbl 0679.20041

[10] F. Levin and G. Rosenberger: “Generators of finite projective linear groups, Part 2”, Results of mathematics, Vol. 17, (1990), pp. 120–127. | Zbl 0703.20047

[11] A.M. Macbeath: “Generators of the linear fractional groups”, Proc. Symp. Pure Math, Vol. 12, (1967), pp. 14–32.

[12] C.H. Sah: “Groups related to compact Riemann surfaces”, Acta Math, Vol. 123, (1969), pp. 13–42. http://dx.doi.org/10.1007/BF02392383 | Zbl 0208.10002

[13] D. Singerman: “Symmetries of Riemann surfaces with large automorphism group”, Math. Ann., Vol. 210, (1974), pp. 17–32. http://dx.doi.org/10.1007/BF01344543 | Zbl 0272.30022

[14] D. Singerman: “On the structure of non-euclidean crystallographic groups”, Proc. Camb. Phil. Soc., Vol. 76, (1974), pp. 233–240. http://dx.doi.org/10.1017/S0305004100048891 | Zbl 0284.20053