On Macbeath-Singerman symmetries of Belyi surfaces with \(\text{PSL} (2,p)\) as a group of automorphisms
type: Item
instance of: scholarly article
title: On Macbeath-Singerman symmetries of Belyi surfaces with \(\text{PSL} (2,p)\) as a group of automorphisms
zbMATH Open document ID: 1074.30040
DOI: 10.2478/BF02476009
author: Ewa Tyszkowska
published in: Central European Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
review text: A Belyi surface is a compact Riemann surface which can be uniformized by a finite index subgroup of a Fuchsian triangle group \(\Gamma\). In that case a finite factor group of \(\Gamma\) acts on a compact Riemann surface as a group of automorphisms. This in mind it is clear that \(\text{PSL}(2,p^n)\), \(p\) prime, acts as a group of automorphisms on a Belyi surface. On the other hand a Belyi surface \(X\) admits a symmetry, and the set of fixed points of a symmetry consists of \(k\), \(1\leq k\leq g+1\), disjoint curves where \(g\) is the genus of \(X\). Here it is shown that \[ k\leq\begin{cases} {3(p-1)\over 4}\quad &\text{if }p\equiv 1(4),\\ {3(p+1)\over 4}\quad &\text{if }p\equiv 3(4)\end{cases} \] for a Belyi surface with \(\text{PSL}(2,p)\), \(p\) an odd prime, as a group of automorphisms and that these bounds are sharp for \(p\equiv 1(8)\) and \(p\equiv 3(8)\), respectively.
reviewed by: Gerhard Rosenberger
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 1978733
zbMATH Keywords: Riemann surface
zbMATH Keywords: automorphism
zbMATH Keywords: symmetry
zbMATH Keywords: ovals
zbMATH Keywords: minimum genus action
zbMATH Keywords: finite projective special linear groups
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.2478/bf02476009
OpenAlex ID: W1985297709
description: scientific article; zbMATH DE number 1978733