Compact Riemann surfaces with large automorphism groups (Q1298064)

Let \(X\) be a compact Riemann surface of genus \(g\geq 2\), having a finite Galois branched covering \(\pi : X \longrightarrow {\mathbb P}^1\). Denote the covering transformation group of \(\pi\) by \(\Aut (\pi)\). Then, \(\Aut (\pi)\) is a subgroup of \(\Aut (X)\), the automorphism group of \(X\). Assuming the order of \(\Aut (\pi)\) is greater than \(4(g-1)\), one can list up the possible branching indices of \(\pi\). In this paper the author divides these branching indices into two classes and shows that \(\Aut (X) =\) \(\Aut (\pi)\) always holds for the indices of one of the classes and not necessarily holds for those of another class.