Bulletin of the
Korean Mathematical Society BKMS

In 1995 it was proved by Gonz\'alez-Diez that the cyclic group generated by a $p$-gonal automorphism of a closed Riemann surface of genus at least two is unique up to conjugation in the full group of conformal automorphisms. Later, in 2008, Gromadzki provided a different and shorter proof of the same fact using the Castelnuovo-Severi theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the Castelnuovo-Severi theorem. This method permits to obtain that the cyclic group generated by a conformal automorphism of order $p$ of a handlebody with a Kleinian structure and quotient the three-ball is unique up to conjugation in the full group of conformal automorphisms.

Keywords: Riemann surfaces, conformal automorphisms, fixed points

MSC numbers: 30F10, 14H37, 14H55