On conjugacy of \(p\)-gonal automorphisms (Q2880481)

A Riemann surface \(S\) of genus \(g \geq 2\) is said to be \(p\)-gonal if there exists an automorphism \(\varphi\) of \(S\) of prime order \(p\) such that \(S/\left<\varphi\right>\) has genus \(0\). Then the cyclic group \(\left<\varphi\right>\) is unique up to conjugation in the group Aut(\(S\)) of conformal automorphisms of \(S\). This was proved by \textit{G. González Diez} [Ann. Mat. Pura Appl. (4) 168, 1--15 (1995; Zbl 0846.30029)], and a simpler proof was obtained by \textit{G. Gromadzki} [Rev. Mat. Complut. 21, No. 1, 83--87 (Zbl 1148.14013)].NEWLINENEWLINEThe author gives here a new proof which is still shorter, by using Castelnuovo-Severi and Sylow theorems. When \(g > (p-1)^2\), by Castelnuovo-Severi the group \(\left<\varphi\right>\) is unique. In case \(g \leq (p-1)^2\), by Sylow theorem, if there are several conjugacy classes \(\left<\varphi\right>\) is a subgroup of an Abelian group of order \(p^2\). A study of the different possibilities on the branch points of \(S \rightarrow S/\left<\varphi\right>\) gives the result.NEWLINENEWLINEThe technique of the proof is then used by the author to prove an analogous result on handlebodies, namely the following.NEWLINENEWLINETheorem 2. Let \(M\) be a handlebody of genus \(g \geq 2\) and \(\varphi \in\) Aut(\(M\)) a conformal automorphism of order \(p\) prime, so that \(M/\left<\varphi\right>\) is a 3-ball. Then the cyclic group \(\left<\varphi\right>\) is unique up to conjugation in Aut(\(M\)).