Lifting the hyperelliptic involution of a Klein surface
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, whilst in the two other cases sufficient conditions are obtained. Then, in Section 5 two kinds of groups are considered. If \(G\) is generated by involutions, then the hyperelliptic involution lifts in all six cases. Finally, if \(G\) is generated by two elements \(a\) and \(b\), \(a\) being an involution, and \(G\) admits an automorphism sending \(a\) onto \(a\) and \(b\) onto \(b^{-1}\), then the hyperelliptic involution lifts in all six cases; for instance the groups \((2,3,r;2)\) and PSL\((2,q)\) satisfy this condition.
scientific article; zbMATH DE number 6946764
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