Francisco-Javier Cirre

On extendability of group actions on compact Riemann surfaces

The question of whether a given group G which acts faithfully on a compact Riemann surface X of genus g > 2 is the full group of automorphisms of X (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group G in terms of a concrete partial presentation for G associated with the relevant branching data, using Singermans list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where G is a non-cyclic abelian group.

On compact Riemann surfaces with dihedral groups of automorphisms

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group DN. The question of extendability of the action of DN is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

Extensions of finite cyclic group actions on non-orientable surfaces

Conditions are derived for the extension of an action of a cyclic group on a compact non-orientable surface to the faithful action of some larger group on the same surface. It is shown that if such a cyclic action is realised by means of a non-maximal NEC signature, then the action always extends. The special case where the full automorphism group is cyclic of the largest possible order (for given genus) is also considered. This extends previous work by the authors for group actions on orientable surfaces. In addition, the smallest algebraic genus of a non-orientable surface on which a given cyclic group acts as the full automorphism group is determined.