Doubles of Klein surfaces (Q2902674)

A Klein surface is a surface with a dianalytic structure. The term Klein surface goes back to Felix Klein, but the modern concept of these surfaces appears in [\textit{N. L. Alling} and \textit{N. Greenleaf}, Foundations of the theory of Klein surfaces. Berlin etc.: Springer (1971; Zbl 0225.30001)]. Klein surfaces are a generalization of Riemann surfaces. A Klein surface can be non-orientable and can have boundary. Riemann surfaces are orientable and unbordered Klein surfaces.NEWLINENEWLINEEvery Klein surface can be represented as a quotient \(X=\mathcal{U}/\Gamma\), where \(\mathcal{U}\) is a simply connected Riemann surface and \(\Gamma\) is a crystallographic group without elliptic elements. The group \(\Gamma\) might have reflections. If the algebraic genus of \(X\) is 1, then \(\mathcal{U}=\mathbb{C}\) and \(\Gamma\) is an Euclidean group. If the algebraic genus of \(X\) is greater than 1, then \(\mathcal{U}=\mathcal{H}\) and \(\Gamma\) is an NEC group. In both cases \(\Gamma\) has a signature of the form \((g;\pm;[\;\;];\{(\;)^k\})\), where \(g\) is the topological genus of \(X\), the sign `+' or `-' appears according to \(X\) is orientable or not, and \(k\geq 0\) is the number of boundary components of \(X\). The group \(\Gamma\) has a collection of canonical generators: \(E=\{e_1,\dots, e_k\}\), \(C=\{c_1,\dots, c_k\}\), and \(A=\{a_1,b_1,\dots,a_g,b_g\}\), (if the sign in the signature is `+') or \(A=\{d_1,\dots,d_g\}\), (if the sign in the signature is `-'), where \(e_i\), \(a_i\) and \(b_i\) are hyperbolic elements, \(c_i\) are reflections and \(d_i\) are glide-reflections. These generators satisfy several canonical relations.NEWLINENEWLINEIn the paper under review doubles of Klein surfaces are studied. Let \(X=\mathcal{U}/\Gamma\) be a Klein surface. A double cover of \(X\) is a quotient of the form \(\mathcal{U}/\Lambda\), where \(\Lambda\) is a subgroup of index 2 in \(\Gamma\). The number of double covers of a given Klein surface, and the number of boundary components of \(\mathcal{U}/\Lambda\) appear in the above quoted book and in the Doctoral Thesis of the second author, respectively. In this paper fresh, short and clear proofs are given by means of NEC groups. Then the authors prove the main result about the orientability character of double covers, which is as follows:NEWLINENEWLINEi) If \(X\) is orientable then \(\mathcal{U}/\Lambda\) is non-orientable if and only if \(\Gamma \setminus \Lambda\) contains both orientation-preserving and orientation-reversing canonical generators of \(\Gamma\).NEWLINENEWLINEii) If \(X\) is non-orientable then \(\mathcal{U}/\Lambda\) is non-orientable if and only if \(\Gamma \setminus \Lambda\) contains both orientation-preserving and orientation-reversing canonical generators of \(\Gamma\) or the group \(\Lambda\) contains any of the glide reflections that are canonical generators of \(\Gamma\).NEWLINENEWLINESince the number of double covers can be a large number when \(g\) and \(k\) grow, the authors study particular doubles that include the most important ones. It is done considering the (standard) epimorphisms from: \(\Gamma\) onto the cyclic group \(C_2\) such that the images of all generators in \(E\) are the same, and similarly for the sets \(C\) and \(A\). It is obtained that if \(k\) is odd, there are three standard epimorphisms, and if \(k\) is even, there are seven standard epimorphisms. Each epimorphism provides a double cover.NEWLINENEWLINENext the authors consider the doubles of the Möbius band as an example. According to the previous result, this surface has three doubles. The geometrical study by means of fundamental regions shows that these doubles are the most important doubles, namely, the complex double, the orienting double and the Schottky double. Afterwards these three doubles are studied for Klein surfaces in general. Finally, the other four doubles (in the case \(k\) even) are studied.