On gonality automorphisms of \(p\)-hyperelliptic Riemann surfaces (Q981976)

Let \(X\) denote a compact Riemann surface of genus \(g\geq 2\). \(X\) is said to be \(p\)-hyperelliptic if \(X\) admits a conformal involution \(\rho\) such that \(X/\langle \rho \rangle\) has genus \(p\). \(X\) is said to be \((q,n)\)-gonal if \(X\) admits a conformal automorphism \(\delta\) of order \(n\) such that \(X/\langle \delta \rangle\) has genus \(q\). In the paper under review, the author provides necessary and sufficient conditions on \(p\) and \(g\) for the existence of a Riemann surface of genus \(g\) admitting a commuting \(p\)-hyperelliptic automorphism \(\rho\) and \((q,n)\)-gonal automorphism \(\delta\) where \(n\) is prime. The author also considers the automorphism group of such a surface and the number of fixed points of \(\delta\). The author finishes by considering the special case where \(\delta\) is central and admits just \(8\) fixed points. The results are obtained using Fuchsian groups, uniformization and utilizing the Riemann-Hurwitz formula. For example, to determine the genus of a surface which admits a commuting \(p\)-hyperelliptic automorphisms and \((q,n)\)-gonal automorphism, the author uses the fact that the group generated by two such automorphisms will be cyclic of order \(2n\). Then, as the author already knows the genera of the quotient spaces and the orders of all fixed points (either \(2\), \(n\) or \(2n\)), they are able to manipulate the Riemann-Hurwitz formula to extract the genus of such a surface. The other results are derived using arguments of a similar theme.