On fixed points of automorphisms of non-orientable unbordered Klein surfaces
Let \(X\) be a compact Riemann surface and \(\varphi\) an automorphism of \(X\). The set of fixed points of \(\varphi\) consists of isolated points or simple closed Jordan curves called ovals. The second case only occurs for anticonformal involutions -- so-called symmetries. Formulae for the numbers of fixed points of an arbitrary automorphism and for the number of ovals of a symmetry are known. In the case of non-orientable surfaces (Klein surfaces) both types of fixed points (isolated and ovals) may occur simultaneously. If the group of automorphisms of \(X\) is cyclic, the corresponding formulae were obtained in [\textit{M. Izquierdo} and \textit{D. Singerman}, Geom. Topol. Monogr. 1, 295-301 (1998; Zbl 0913.20019)]. In the paper under review, the author finds similar formulae when the group of automorphisms of \(X\) is arbitrary. Since the considered surfaces are compact non-orientable surfaces, non-Euclidean crystallographic groups, NEC groups in short, play an essential role. Let \(X=\mathcal{H}/\Gamma\) be a non-orientable unbordered Klein surface, where \(\mathcal{H}\) denotes the hyperbolic plane and \(\Gamma\) is a surface NEC group. Let \(G\) be a group of automorphisms of \(X\). Then there exists another NEC group \(\Lambda\) and an epimorphism \(\theta:\Lambda\rightarrow G\) such that ker\((\theta)=\Gamma\). In the paper, the number of isolated fixed points of \(\varphi\in G\) is given in terms of the order of the normalizer of \(<\varphi>\) and some proper periods and link-periods in the signature of \(\Lambda\) corresponding to some canonical generators. The number of ovals fixed by an involution \(\sigma\in G\) is given by \(\sum[C(G,\theta(c)):\theta(C(\Lambda,c))]\), where \(C\) denotes the centralizer and the sum is taken over non-conjugate reflections of \(\Lambda\), whose images under \(\theta\) are conjugate to \(\sigma\) in \(G\). At the end of the paper, a couple of examples are developed, in which the topological type of the sets of points fixed by automorphisms acting on extremal Klein surfaces are determined.
automorphism of Riemann and Klein surfaces
scientific article; zbMATH DE number 5500102