On symmetric representations of groups of automorphism of bordered Klein surfaces
type: Item
instance of: scholarly article
title: On symmetric representations of groups of automorphism of bordered Klein surfaces
zbMATH Open document ID: 1264.30029
author: Czesław Bagiński
author: G. Gromadzki
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2013-01-30T00:00:00Z
review text: A Klein surface is a compact topological surface with a dianalytic structure. A bordered Klein surface \(X\) of algebraic genus \(g\geq2\) can be represented as a quotient \(X=\mathcal{H}/\Gamma\), where \(\mathcal{H}\) denotes the hyperbolic plane and \(\Gamma\) is an non-euclidean crystallographic group (NEC group, in short). The group \(\Gamma\) has a signature \((g',\pm,[-],\{(-)^k \})\), where \(k>0\) is the number of boundary components of \(X\), the algebraic genus of \(X\) is \(g=\epsilon g'+k-1\), where \(\epsilon=2\) if \(X\) is an orientable surface (sign `\(+\)' in the above signature) or \(\epsilon =1\) if \(X\) is a non-orientable surface (sign `\(-\)'). A finite group \(G\) is a group of automorphisms of \(X=\mathcal{H}/\Gamma\) if there exists an NEC group \(\Lambda\) and an epimorphism \(\theta: \Lambda \rightarrow G\) such that \(\Gamma= \ker\theta\) and \(G=\Lambda/\Gamma\). An automorphism of \(X\) induces a permutation of its boundary components, \(\mathcal{O}_1, \dots, \mathcal{O}_k\). In the paper under review, the authors study the corresponding groups of representations of groups of automorphisms of \(X\) in the corresponding finite symmetric groups \(S_k\). Let \(\rho_X:G\rightarrow S_k\) be given by \(\rho_X(g)(i)=j \Leftrightarrow g(\mathcal{O}_i)=\mathcal{O}_j\). This representation is called geometric. The authors prove that \(\rho_X\) is a homomorphism whose kernel is either cyclic or dihedral. Next, they study this kernel. Given a set of canonical generators of \(\Lambda\), let \(\mathcal{C}\) be the set of its canonical reflections. Let \(\{c_i : i\in I\}\) be a maximal subset of \(\mathcal{C}\) consisting of the pairwise non-conjugate reflections, mapped by \(\theta\) into the unit. Then it is proved that \[ \ker\rho_X=\bigcap_{g\in G} \bigcap_{i\in I}g\big(\theta(C(\Lambda,c_i))\big)g^{-1} \] where \(C\) denotes the centralizer group. In the main result, the authors prove that an abstract representation \(\rho:G\rightarrow S_k\) is…
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 6132872
zbMATH Keywords: Klein surfaces
zbMATH Keywords: bordered Klein surfaces
zbMATH Keywords: automorphisms of Klein surfaces
zbMATH Keywords: NEC-groups
reviewed by: Ernesto Martínez
MaRDI profile type: Publication
full work available at URL: s13398-012-0062-x
OpenAlex ID: W2051019317
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DOI: 10.1007/S13398-012-0062-X
description: scientific article; zbMATH DE number 6132872