Automorphism groups of the real projective plane with holes and their conjugacy classes within its mapping class group
type: Item
instance of: scholarly article
title: Automorphism groups of the real projective plane with holes and their conjugacy classes within its mapping class group
zbMATH Open document ID: 1068.30034
published in: Mathematische Annalen
publication date: 2005-06-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 2172736
zbMATH Keywords: family of automorphism groups of compact non-orientable Klein surfaces with boundary components
zbMATH Keywords: real algebraic curves
zbMATH Keywords: ovals
zbMATH Keywords: finite subgroups of mapping class groups of a non-orientable surface
zbMATH Keywords: conjugacy classes
zbMATH Keywords: representatives
zbMATH Keywords: non-equivalent marked signatures
author: Emilio Bujalance
author: Francisco Javier Cirre
author: Jose Manuel Gamboa
MaRDI profile type: Publication
full work available at URL: s00208-004-0621-3
OpenAlex ID: W2146724810
cites work: Q3282112
cites work: Foundations of the theory of Klein surfaces
cites work: Q5640448
cites work: Automorphism groups of compact planar Klein surfaces
cites work: Cyclic trigonal Klein surfaces
cites work: Automorphism groups of compact bordered Klein surfaces. A combinatorial approach
cites work: The full automorphism groups of hyperelliptic Riemann surfaces
cites work: Automorphism groups of real algebraic curves which are double covers of the real projective plane
cites work: The moduli space of real algebraic curves of genus 2.
cites work: Q5622541
cites work: The Nielsen realization problem
cites work: The Word Problem and Consequences for the Braid Groups and Mapping Class Groups of the 2-Sphere
cites work: On Branch Loci in Teichmuller Space
cites work: The Classification of Non-Euclidean Plane Crystallographic Groups
cites work: Spaces of Subgroups and Teichmüller Space
cites work: On Mapping-Class Groups and Teichmüller Spaces
cites work: Q3033885
cites work: Notes on conformal mappings of a Riemann surface onto itself
cites work: Q5835829
DOI: 10.1007/S00208-004-0621-3
description: scientific article; zbMATH DE number 2172736