Automorphism groups of compact Riemann surfaces of genus five (Q921170)

Let X be a compact Riemann surface of genus \(\geq 2\), and let AG be a group of automorphisms of X. Since AG acts on the vector space of holomorphic differentials on X, it can be represented as a subgroup R(X,AG) of \(GL_ g({\mathbb{C}})\). It arises a problem: Determine all subgroups of \(GL_ g({\mathbb{C}})\) which are conjugate to R(X,AG) for some X and some AG. For \(g=2\), this problem was already solved by I. Kuribayashi and for \(g=3,4\) it was solved by I. Kuribayashi and the first author. In this paper, the authors solve it for \(g=5\).