Non-orientable Klein Surfaces Research Articles

Automorphism groups of bordered non-orientable Klein surfaces of topological genus 4

Automorphism groups of bordered non-orientable Klein surfaces of topological genus 4

Groups of automorphisms of bordered non-orientable Klein surfaces of topological genus 3

In this paper we study the automorphism groups G of compact non-orientable Klein surfaces of topological genus 3 with k>0 boundary components. We obtain that the group G is one of the following: C2, C3, C4, C6, C2×C2, D3, D4 and D6. Besides, for each k>0 there exists such a Klein surface admitting each of these groups as an automorphism group, excepting that C6 and D6 do not act on surfaces with k≡1mod3.Moreover, if G is an automorphism group of a bordered non-orientable Klein surface of topological genus 3, with k boundary components, then it is the full automorphism group of some such surface excepting C3 for k=1 and 2, C4 for k=1, and C6 for k=2 and 5.

On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus [Formula: see text] with one boundary component is connected and in the case of non-orientable Klein surfaces it has [Formula: see text] components, if [Formula: see text] is odd, and [Formula: see text] components for even [Formula: see text]. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by [Formula: see text]. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with [Formula: see text] is finite. The value of [Formula: see text] is known when G belongs to certain families of groups. In particular, if o(G) < 32, [Formula: see text] is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.

On the set of fixed points of automorphisms of bordered Klein surfaces

The nature of the set of points fixed by automorphisms of Riemann or unbordered nonorientable Klein surfaces as well as quantitative formulae for them were found by Macbeath, Izquierdo, Singerman and Gromadzki in a series of papers. The possible set of points fixed by involutions of bordered Klein surfaces has been found by Bujalance, Costa, Natanzon and Singerman who showed that it consists of isolated fixed points, ovals and chains of arcs. They classified involutions of such surfaces, up to topological conjugacy in these terms. Here we give formulae for the number of elements of each type, also for non-involutory automorphisms, in terms of the topological type of the action of the group of dianalytic automorphisms. Finally we give some illustrative examples concerning bordered Klein surfaces with large groups of automorphisms already considered by May and Bujalance.

The symmetric cross-cap number of the groups Cm × Dn

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by $\tilde{\sigma}(G)$. This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus.The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm × Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.

Dynamics of dianalytic transformations of Klein surfaces

This paper is an introduction to dynamics of dianalytic self-maps of nonorientable Klein surfaces. The main theorem asserts that dianalytic dynamics on Klein surfaces can be canonically reduced to dynamics of some classes of analytic self-maps on their orientable double covers. A complete list of those maps is given in the case where the respective Klein surfaces are the real projective plane, the pointed real projective plane and the Klein bottle.

Mapping class group of a non-orientable surface and moduli space of Klein surfaces

As for Riemann surfaces, the moduli space of closed non-orientable Klein surfaces of genus g can be defined as the orbit space of the Teichmüller space T g by the mapping class group Mod g of a closed non-orientable surface. Using the set of generators given by Birman and Chillingworth, we prove that the latter group is generated by involutions. We conclude, using the Armstrong's result, that the moduli space is simply-connected. To cite this article: B. Szepietowski, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1053–1056.

On non-orientable Riemann surfaces with 2p or 2p + 2 automorphisms

It is known that the maximal order of a cyclic group of automorphisms admitted by a Klein surface or real algebraic curve of algebraic genus p is 2p or 2(p + 1), depending on whether p is odd or even. In this paper, we classify the automorphism groups of all non-orientable Klein surfaces, without boundary, which admit an automorphism group of order 2p, or 2(p + 1). We determine that the automorphism groups are cyclic precisely when the surfaces are hyperelliptic. Defining equations for all but one family of these Klein surfaces are given.

Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary

Cyclic groups of automorphisms of compact nonorientable Klein surfaces without boundary

Analytic and harmonic obstruction on nonorientable Klein surfaces

Analytic and harmonic obstruction on nonorientable Klein surfaces