Lifting the hyperelliptic involution of a Klein surface
type: Item
instance of: scholarly article
title: Lifting the hyperelliptic involution of a Klein surface
zbMATH Open document ID: 1409.30042
author: Francisco Javier Cirre
author: Emilio Bujalance
author: Peter Turbek
published in: Revista Matemática Complutense
publication date: 2018-10-04T00:00:00Z
review text: A Klein surface \(X\) is a connected surface endowed with an atlas whose transition functions are analytic or antianalytic. This fact allows \(X\) to be orientable or non-orientable, and bordered or unbordered. Orientable unbordered Klein surfaces are Riemann surfaces, and if \(X\) is not a Riemann surface, it is a proper Klein surface. The present paper deals with unbranched coverings \(\pi : X \rightarrow X'\) between proper Klein surfaces. The main tool used for the study of these surfaces is their uniformization by means of NEC groups, according to which a surface \(X\) can be expressed as \(U/\Gamma\) where \(U\) is the hyperbolic plane and \(\Gamma\) is an NEC group. Then, if \(G\) is the group of covering transformations of the covering, \(G = \Gamma' / \Gamma\) where \(\Gamma'\) is another NEC group containing \(\Gamma\) as a normal subgroup. The authors are interested in the case where \(X'\) is hyperelliptic, that is to say, \(X'\) admits an involution, say \(h_{X'}\), such that \(X' / \langle h_{X'}\rangle\) has algebraic genus \(0\). The goal of the paper is to obtain the conditions that \(G\) must satisfy in order to get that the hyperelliptic involution of \(X\), \(h_{X'}\), lifts to an automorphism \(\phi\) of \(X\), that is to say, there exists \(\phi\) such that \(\pi \phi = h_{X'} \pi\). The study splits naturally into six cases depending on the topological types of \(X'\) which were obtained by \textit{E. Bujalance} et al. in [Lect. Notes Math. 1398, 43--51 (1989; Zbl 0682.20035)] for the unbordered case, and in [Q. J. Math., Oxf. II. Ser. 36, 141--157 (1985; Zbl 0573.14010)] in the bordered case. The result is given in Theorem 3.5 which collects the details of Section 3, and expresses the necessary and sufficient conditions that \(G\) must satisfy. The last two sections apply Theorem 3.5 to some special cases. Firstly, in Section 4, if \(G\) is abelian, it is proved that in four of the topological types the hyperelliptic involution always lifts, …
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 6946764
zbMATH Keywords: Klein surface
zbMATH Keywords: normal covering
zbMATH Keywords: automorphism group
zbMATH Keywords: lifting automorphisms
reviewed by: José Javier Etayo
MaRDI profile type: Publication
full work available at URL: s13163-018-0264-7
OpenAlex ID: W2800798796
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DOI: 10.1007/S13163-018-0264-7
description: scientific article; zbMATH DE number 6946764