Emilio Bujalance

Automorphism Groups of Compact Bordered Klein Surfaces

Preliminary results.- Klein surfaces as orbit spaces of NEC groups.- Normal NEC subgroups of NEC groups.- Cyclic groups of automorphisms of compact Klein surfaces.- Klein surfaces with groups of automorphisms in prescribed families.- The automorphism group of compact Klein surfaces with one boundary component.- The automorphism group of hyperelliptic compact Klein surfaces with boundary.

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g.

On Cyclic Groups of Automorphisms of Riemann Surfaces

The question of extendability of the action of a cyclic group of automorphisms of a compact Riemann surface is considered. Particular attention is paid to those cases corresponding to Singermans list of Fuchsian groups which are not nitely-maximal, and more generally to cases involving a Fuchsian triangle group. The results provide partial answers to the question of which cyclic groups are the full automorphism group of some Riemann surface of given genus g > 1.

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.

Involutions of compact Klein surfaces

Let X be a compact Klein surface of topological genus g and k bounda ry components and let tp: X-+X be a dianalytic involution. We are interested in determining q0 up to topological conjugacy by a finite number of invariants mainly connected with Fix(q~), the fixed point set of qo. We shall also investigate corresponding inclusions between non-Eucl idean crystal lographic groups and use these to consider the subspaces of Teichmfiller space of Klein surfaces admitting involutions. As we shall see, Fix(cp) consists of (a) a finite number of isolated fixed points, (b) a finite number of simple closed curves. By analogy with the case of plane algebraic curves we shall call a fixed simple closed curve an oval. Ovals will be called twisted or untwisted according to whether they have M6bius band or annular ne ighbourhoods respectively. Of course, twisted ovals can only occur on non-orientable surfaces.

Symmetries of Riemann surfaces on which PSL(2, q) acts as a Hurwitz automorphism group

Let X be a compact Riemann surface and Aut(X) be its automorphism group. An automorphism of order 2 reversing the orientation is called a symmetry. The authors together with D. Singerman have been working on symmetries of Riemann surfaces in the last decade. In this paper, the symmetry type St(X) of X is defined as an unordered list of species of conjugacy classes of symmetries of X, and for a class of particular surfaces, St(X) is found. This class consists of Riemann surfaces on which PSL(2, q) acts as a Hurwitz group. An algorithm to calculate the symmetry type of this class is provided.

Pseudo-real Riemann surfaces and chiral regular maps

A Riemann surface is called pseudo-real if it admits anticonformal automorphisms but no anticonformal involution. In this paper, we study general properties of the automorphism groups of such surfaces and their uniformizing NEC groups. In particular, we prove that there exist pseudo-real Riemann surfaces of every possible genus g > 2. We also study pseudo-real surfaces of genus 2 and 3. Further, we establish a connection between pseudo-real surfaces with maximal automorphism group, and chiral 3-valent regular maps, and use this to show there exist such surfaces for infinitely many genera, by exhibiting infinite families of chiral regular maps of type {3, k} for all k > 7.

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.

A characterization of q-hyperelliptic compact planar klein surfaces

A compact Klein surface X is called q-hyperelliptic if there is an involution

Topological Types Of P-Hyperelliptic Real Algebraic-Curves

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