On Macbeath-Singerman symmetries of Belyi surfaces with \(\text{PSL} (2,p)\) as a group of automorphisms
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.
scientific article; zbMATH DE number 1978733