On prime Galois coverings of the Riemann sphere (Q1913385)

The author considers cyclic groups of prime order \(p\) acting as group of conformal automorphisms on a closed Riemann surface, so that the quotient has genus zero. If \(p= 2\) it is well-known that such a group is unique (generated by the hyperelliptic involution). If \(p\geq 3\), the uniqueness is not longer true. The author shows that the group is unique up to conjugation by conformal automorphism of the Riemann surface. A typical example is the Klein surface of genus three admitting a group of conformal automorphisms of order 168. In this Riemann surface there are different cyclic subgroups of order 7. The author investigates some consequences of the above result at the level of Teichmüller and moduli spaces.