On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms |
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Authors: | Ewa Tyszkowska |
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Institution: | (1) Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland |
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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 |
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Keywords: | Riemann surface automorphism symmetry ovals minimum genus action finite projective special linear groups MSC (2000) Primary 30F20 30F50 Secondary 14H37 20H30 20H10 |
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