A remark on \((p,n)\)-gonal quasiplatonic Riemann surfaces (Q715198)

A Riemann surface \(S\) of genus \(g \geq 2\) is said to be \((p,n)\)-gonal if there exists an automorphism \(\tau\) of \(S\) of order \(p\) such that \(S/\left<\tau\right>\) has genus \(n\). If \(p\) is prime and \(g > 2pn + (p-1)^2\), \(S\) is called strongly \((p,n)\)-gonal. In these last conditions, the Castelnuovo-Severi theorem implies that the \((p,n)\)-gonal group \(\left<\tau\right>\) is unique in Aut\((S)\). The surface is called quasiplatonic if \(S/\mathrm{Aut}(S)\) has signature \((0;a,b,c)\). The quasiplatonic \((p,n)\)-gonal surfaces were studied by \textit{G. Gromadzki, A. Weaver} and \textit{A. Wootton} [Geom. Dedicata 149, 1--14 (2010; Zbl 1213.14068)]. The paper under review comes from this quoted article, and is devoted to prove that the number of quasiplatonic \((p,n)\)-gonal surfaces is finite up to conformal equivalence for every prime \(p\), and \(n > 1\). Examples are also given of \((p,0)\)- and \((p,1)\)-gonal quasiplatonic surfaces.