, Volume 91, Issue 6, pp 536-543
Date: 05 Dec 2008

Applications of a theorem of Singerman about Fuchsian groups

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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.

The first author is supported in part by MTM2005-01637, the second author was supported in part by SNFS grant number PBEL2-106180.
Received: 4 April 2008