Groups of automorphisms of cyclic trigonal Riemann surfaces (Q731908)

A compact Riemann surface \(X\) is said to be cyclic trigonal if it admits an automorphism \(\rho\) of order three such that the quotient space \(X/\langle \rho \rangle\) has genus zero. It is a three sheeted covering of the Riemann sphere and the sheets are permuted cyclically. The group \(\langle \rho \rangle\) is called a trigonality automorphism group, and it is unique if \(X\) has genus \(g\geq 5\). In this paper the authors present the complete list of groups \(G\) which act as a group of automorphisms of a cyclic trigonal Riemann surface of genus \(g\geq 5\) and which contain the trigonality automorphism group. The authors also determine all ramification types of the covering \(X\to X/G\) for each group \(G\) and show that different ramification types can arise even if \(G\) is an abstract group. The authors obtain these results by a careful use of the combinatorial theory of Fuchsian groups.