Automorphism groups of the real projective plane with holes and their conjugacy classes within its mapping class group
The first goal in this paper is the computation of the family Aut\(_{(g,g,1)}\) of full automorphism groups of all compact non-orientable Klein surfaces of algebraic genus \(g\) with \(g\) boundary components. Under the well known functorial coequivalence between compact Klein surfaces and (smooth, irreducible, projective) real algebraic curves, these Klein surfaces correspond to real algebraic curves of genus \(g\) with \(g\) connected components (or ovals). These curves are usually referred to as \((M-1)\)-curves since, by Harnack's Theorem, they have the second largest number of ovals that a genus \(g\) curve may admit. For each \(g\geq 2\), the complete list of groups of birational automorphisms of \((M -1)\)-curves is given. For later purposes, the authors determine all groups acting as full automorphism groups of such curves. They are isomorphic to finite groups of isometrics of the Riemann sphere. In the second part of this paper, the authors turn the attention to mapping class groups Map\(_{(g,k,\varepsilon)}\) of homotopy classes of selfhomeomorphisms of the surface \(S_{(g,k,\varepsilon)}\) of topological type \((g,k,\varepsilon)\). Very few results are known on the mapping class group of a non-orientable surface, in contrast with the classical Map\(_g\). Here the authors are interested in the conjugacy classes of the finite subgroups of Map\(_{(g,k,1)}\). These finite subgroups can be realized as automorphism groups of Klein surfaces of topological type \((g,g, 1)\). Their conjugacy classes within Map\(_{(g,g,1)}\) are determined, and a representative of each of them and the subsequent list of marked signatures are found.
family of automorphism groups of compact non-orientable Klein surfaces with boundary components
finite subgroups of mapping class groups of a non-orientable surface
scientific article; zbMATH DE number 2172736
scientific article