Finite group actions on bordered surfaces of small genus
type: Item
instance of: scholarly article
title: Finite group actions on bordered surfaces of small genus
zbMATH Open document ID: 1209.30016
published in: Journal of Pure and Applied Algebra
publication date: 2010-08-27T00:00:00Z
review text: A compact bordered Klein surface may be seen as the quotient of an unbordered orientable surface under the action of a symmetry with fixed points. A bordered surface may be endowed with a dianalytic structure and is called a Klein surface. Most Klein surfaces admit only the trivial automorphism. Surfaces with algebraic genus \(p>2\) with a non-trivial automorphism constitute the singular locus of the moduli space of compact Klein surfaces of a given topological type. Many results about the group of automorphisms of bordered Klein surfaces are known, but the list of all finite groups of automorphisms of these surfaces are known only for algebraic genus \(2\) and \(3\), and they were obtained in [\textit{E. Bujalance, J. M. Gamboa}, Arch. Math. 42, 229--237 (1984; Zbl 0522.14016)] and [\textit{E. Bujalance, J. J. Etayo, J. M. Gamboa}, Proc. Japan Acad. Ser. A. 62, 40--42 (1986; Zbl 0594.14019)], respectively. A very interesting problem is to determine for each group of automorphisms of a bordered surface the inequivalent topological group actions. This is a finer classification and it is necessary for the study of the singular locus in the moduli space. This space can be viewed as the quotient of the Teichmüller space under the action of the mapping class group. The conjugacy classes of finite subgroups of the mapping class group are in one-to-one correspondence with the topological classes of finite groups actions. In the paper under review, the authors determine, up to topological equivalence, all finite group actions on compact bordered surfaces of algebraic genus \(p\) for \(2\leq p\leq 6\). These surfaces can be orientable or non-orientable, and they are topologically classified by means of three parameters, namely, the algebraic genus \(p\), the number of boundary components \(k\), and its orientability. The triple \((p,k,\epsilon)\), where \(\epsilon=+\) or \(\epsilon=-\) according to the surface is orientable or not, is called the topological type of the surfa…
reviewed by: Ernesto Martínez
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 5775711
zbMATH Keywords: Finite group
zbMATH Keywords: Klein surfaces
zbMATH Keywords: \(NEC\) groups
author: Emilio Bujalance
author: Francisco Javier Cirre
author: Marston D. E. Conder
author: Błazej Szepietowski
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.02.020
OpenAlex ID: W2085200573
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DOI: 10.1016/J.JPAA.2010.02.020
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description: scientific article; zbMATH DE number 5775711