Topological classification of conformal actions on p-hyperelliptic and (q,n)-gonal Riemann surfaces

A compact Riemann surface \(X\) of genus \(g \gt 1\) is said to be \(p\)-hyperelliptic if \(X\) admits a conformal involution \(\rho\) for which \(X / \rho\) has genus \(p\). A conformal automorphism \(\delta\) of prime order \(n\) such that \(X / \delta\) has genus \(q\) is called a \((q,n)\)-gonal automorphism. Here we study conformal actions on \(p\)-hyperelliptic Riemann surface with \((q,n)\)-gonal automorphism