Applications of a theorem of Singerman about Fuchsian groups

Abstract.

Assume that we have a (compact) Riemann surface S, of genus greater than 2, with $S = {\mathbb{D}}/ \Gamma$, where ${\mathbb{D}}$ is the complex unit disc and Γ is a surface Fuchsian group. Let us further consider that S has an automorphism group G in such a way that the orbifold S/G is isomorphic to ${\mathbb{D}}/\Gamma^\prime$ where $\Gamma^\prime$ is a Fuchsian group such that $\Gamma \vartriangleleft \Gamma^\prime$ and $\Gamma^\prime$ has signature σ appearing in the list of non-finitely maximal signatures of Fuchsian groups of Theorems 1 and 2 in [6]. We establish an algebraic condition for G such that if G satisfies such a condition then the group of automorphisms of S is strictly greater than G, i.e., the surface S is more symmetric that we are supposing. In these cases, we establish analytic information on S from topological and algebraic conditions.