Representing \(PSL_ 2(p)\) on a Riemann surface of least genus
For any finite group G denote by genus (G) the least genus of all Riemann surfaces S for which there exists an effective conformal action \(G\times S\to S\). We note that genus(G) has also been called the symmetric genus of G in the literature on combinatorial theory. This notion is closely related to the Cayley genus of a group but did not coincide with it. For details see [\textit{T. Tucker}, J. Comb. Theory, Ser. B 34, 82-92 (1983; Zbl 0521.05027)]. The purpose of the present paper is to determine the genus for the simple groups PSL\({}_ 2(p)\), where p is a prime \(\geq 5\). One of the most important steps in the proof of the main theorem of this paper is the following result. Let S be a surface of minimal genus for \(G=PSL_ 2(p)\). Then the orbit manifold S/G is the two-sphere \(S^ 2\) and the quotient map \(S\to S^ 2=S/G\) is a branched covering with exactly 3 branch points.
scientific article; zbMATH DE number 3955357