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

Open Mathematics (2003)

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

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topEwa 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] 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
- [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 Zbl0847.30026
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- [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
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- [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 Zbl0208.10002
- [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 Zbl0272.30022
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