Automorphisms of non-cyclic $p$-gonal Riemann surfaces

In this paper we prove that the order of a holomorphic automorphism of a non-cyclic $p$-gonal compact Riemann surface $S$ of genus $g>(p-1)^2$ is bounded above by $2(g+p-1)$. We also show that this maximal order is attained for infinitely many genera. This generalises the similar result for the particular case $p=3$ recently obtained by Costa-Izquierdo. Moreover, we also observe that the full group of holomorphic automorphisms of $S$ is either the trivial group or is a finite cyclic group or a dihedral group or one of the Platonic groups $\mathcal A_4$, $\mathcal A_5$ and $\Sigma_4$. Examples in each case are also provided. If $S$ admits a holomorphic automorphism of order $2(g+p-1)$, then its full group of automorphisms is the cyclic group generated by it and every $p$-gonal map of $S$ is necessarily simply branched.
Finally, we note that each pair $(S,\pi)$, where $S$ is a non-cyclic $p$-gonal Riemann surface and $\pi$ is a $p$-gonal map, can be defined over its field of moduli. Also, if the group of automorphisms of $S$ is different from a non-trivial cyclic group and $g>(p-1)^2$, then $S$ can be also be defined over its field of moduli.