Automorphisms of ramified coverings of Riemann surfaces
Let \(M\), \(N\) be Riemann surfaces, and \(\pi: M\to N\) be an unlimited, ramified, finite-sheeted covering map. We say that \((M, \pi, N)\) is of excluded type if \(N= \mathbb{C}\) and the image set of ramification points of \(\pi\) consists of one point on \(N\). The main result of the paper is the following. Suppose that the above-mentioned covering map \(\pi: M\to N\) is of non-excluded type. Then there exist a Riemann surface \(S\) and unlimited covering maps \(\sigma: M\to S\) and \(q: S\to N\) such that \(\pi= q\circ \sigma\), where \(q\) is unramified, and for any \(T\in \Aut(M)\) fixing all of the ramification points of \(\pi\) there exists a \(T_*\in \Aut(S)\) which satisfies \(\sigma\circ T= T_*\circ \sigma\). The following corollaries are of special interest. Corollary 1. Let \(\pi: M\to N\) be an unlimited, ramified, \(l\)-sheeted covering, and of non-excluded type. Suppose that \(l\) is a prime number. Then for any arbitrary \(T\in \Aut(M)\) fixing all of the ramification points of \(\pi\), there exists a \(T_*\in \Aut(N)\) which satisfies \(\pi\circ T= T_*\circ \pi\). Corollary 2. Let \(\pi: M\to N\) be an unlimited, ramified, finite-sheeted covering, and of non-excluded type. Suppose that \(N\) is simply connected. Then for an arbitrary \(T\in \Aut(M)\) fixing all of the ramification points of \(\pi\), there exists a \(T_*\in \Aut(N)\) which satisfies \(\pi\circ T= T_*\circ \pi\).
scientific article; zbMATH DE number 840319