On conjugacy of \(p\)-gonal automorphisms of Riemann surfaces
A compact Riemann surface \(X\) of genus \(g\geq 2\) is said to be cyclic \(p\)-gonal if there exists an automorphism \(\phi\) of order \(p\) such that the orbit space \(X/\langle \phi \rangle\) has genus \(0\). Such an automorphism is called a \(p\)-gonal automorphism. Let \(A\) denote the full automorphism group of \(X\). The Castelnuovo-Severi theorem [\textit{G. Castelnuovo}, Rom. Acc. L. Rend. (5) 15, 337--344 (1906; JFM 37.0600.01)] asserts that for \(g>(p-1)^2\), the group generated by a \(p\)-gonal automorphism is unique in \(A\). For \(g\leq (p-1)^2\), this is no longer the case, and there are examples of surfaces for which the \(p\)-gonal automorphism is not unique such as Klein's genus \(3\) surface. Though the \(p\)-gonal automorphism is not unique for \(g\leq (p-1)^2\), in \textit{G. Gonzalez-Diez} [Ann. Mat. Pura Appl. (4) 168, 1--15 (1995; Zbl 0846.30029)], it was shown that there is a unique conjugacy class of \(p\)-gonal automorphisms in \(A\) for any \(g\). In the paper under review, the author provides a new proof for this result using techniques which may be of use for other similar problems. The proof presented is through contradiction. Suppose that \(a\) and \(b\) are two \(p\)-gonal automorphisms which are not conjugate in \(A\) but lie in the same Sylow subgroup of \(A\) and let \(G=\langle a,b\rangle\). Counting the fixed points of the action of \(G\) on \(X\) and utilizing the Riemann-Hurwitz formula, the author is able to first show that \(X/G\) has genus \(0\), \(| G| =p^2\) and the map \(X\rightarrow X/G\) is branched over three or four points. The author then applies results of \textit{E. Bujalance, F. J. Cirre} and \textit{M. Conder} [Trans. Am. Math. Soc. 355, No. 4, 1537--1557 (2003; Zbl 1019.20018)] to show that in both of these cases, there exists an element of \(A\) which conjugates \(b\) to a power of \(a\) contradicting the assumption that \(a\) and \(b\) are not conjugate.
scientific article; zbMATH DE number 5289681