Maximal group actions on compact oriented surfaces
Let \(p\) be a prime integer and let \(C_{p}\) be the cyclic group of order \(p\). It is well known the existence of closed Riemann surfaces of genus at least two on which \(C_{p}\) acts as a group of conformal automorphisms. Assume \(S\) is such a closed Riemann surface of genus \(\sigma \geq 2\) and that \(p \geq 5\). The quotient orbifold \(S/C_{p}\) has genus \(h\) and exactly \(r \neq 1\) cone points, each one of order \(p\); its signature is then defined by \((h;p^{r})\). In the paper under review, it is observed that: (i) if \(h \leq (p-5)/2\) and \(r>p+7\), then \(\text{Aut}(S)=C_{p}\), and (ii) if \(h \geq (p-3)/2\) and \(r \geq 3\), then there is a (possible different) closed Riemann surface \(R\) of same genus \(\sigma\) admitting also a conformal action of \(C_{p}\) with quotient orbifold \(R/C_{p}\) of the same signature \((h;p^{r})\) and \(C_{p} \neq \text{Aut}(R)\). The arguments are given by studying the possible generating vectors for \(C_{p}\) and the use of the Riemann-Hurwitz formula. At the end of the paper, two explicit examples are provided. The first one corresponds to the two possible actions of \(C_{7}\) in genus three, in both cases the group of conformal automorphisms being bigger, and the second one is a \(C_{7}\) action of genus nine which turns out to be the full group of conformal automorphisms.
scientific article; zbMATH DE number 6668095