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

Ewa Tyszkowska

Open Mathematics (2003)

  • Volume: 1, Issue: 2, page 208-220
  • ISSN: 2391-5455

Abstract

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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

How to cite

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Ewa Tyszkowska. "On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms." Open Mathematics 1.2 (2003): 208-220. <http://eudml.org/doc/268693>.

@article{EwaTyszkowska2003,
abstract = {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},
author = {Ewa Tyszkowska},
journal = {Open Mathematics},
keywords = {Riemann surface; automorphism; symmetry; ovals; minimum genus action; finite projective special linear groups; MSC (2000); Primary; 30F20; 30F50; Secondary; 14H37; 20H30; 20H10},
language = {eng},
number = {2},
pages = {208-220},
title = {On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms},
url = {http://eudml.org/doc/268693},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Ewa Tyszkowska
TI - On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms
JO - Open Mathematics
PY - 2003
VL - 1
IS - 2
SP - 208
EP - 220
AB - 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
LA - eng
KW - Riemann surface; automorphism; symmetry; ovals; minimum genus action; finite projective special linear groups; MSC (2000); Primary; 30F20; 30F50; Secondary; 14H37; 20H30; 20H10
UR - http://eudml.org/doc/268693
ER -

References

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  2. [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. Zbl0709.14021
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  6. [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 Zbl0613.57019
  7. [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 Zbl0885.14026
  8. [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. [9] U. Langer and G. Rosenberger: “Erzeugende endlicher projectiver linearer Gruppen”, Result in Mathematics, Vol. 15, (1989), pp. 119–148. Zbl0679.20041
  10. [10] F. Levin and G. Rosenberger: “Generators of finite projective linear groups, Part 2”, Results of mathematics, Vol. 17, (1990), pp. 120–127. Zbl0703.20047
  11. [11] A.M. Macbeath: “Generators of the linear fractional groups”, Proc. Symp. Pure Math, Vol. 12, (1967), pp. 14–32. 
  12. [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 Zbl0208.10002
  13. [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 Zbl0272.30022
  14. [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 Zbl0284.20053

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