On s -extremal Riemann surfaces of even genus
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On s-extremal Riemann surfaces of even genus Ewa Kozłowska-Walania1 Received: 23 February 2020 / Accepted: 6 November 2020 © The Author(s) 2020
Abstract We consider Riemann surfaces of even genus g with the action of the group Dn × Z2 , with n even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called s-extremal. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an s-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called extremal. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not. Keywords Riemann surface · Symmetry of a Riemann surface · Real form · Automorphisms of Riemann surface · Fuchsian groups · Riemann uniformization theorem · Separating symmetry Mathematics Subject Classification Primary 30F99 · 14H37; Secondary 20F
1 Introduction All Riemann surfaces in this paper are compact. A symmetry of a Riemann surface X = H/ of genus g ≥ 2, where is a Fuchsian surface group, is just an antiholomorphic involution σ ∈ G = Aut± (X ). The set of points fixed by σ consists of no more than g +1 disjoint simple closed curves called ovals, see Harnack [10]. If the set X \Fix(σ ) is disconnected, then we call σ to be separating and we call it non-separating in the other case. In addition, we define the topological type of σ to be a symbol ±k, where
Supported by Polish National Sciences Center by the Grant NCN MINIATURA 3 DEC-2019/03/X/ST1/01239.
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Ewa Kozłowska-Walania [email protected] Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gda´nsk, Wita Stwosza 57, 80-952 Gda´nsk, Poland
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E. Kozłowska-Walania
k ≥ 0 denotes the number of ovals of σ , and the sign depends on the separability of σ : + for separating, − for a non-separating symmetry. The starting point for this paper is the results of Gromadzki and Izquierdo from [6,7], where they prove that the number of non-isomorphic Klein surfaces (X , σ ) for a fixed compact Riemann surface X of even genus is at most 4, and this bound is attained only for the automorphism group of the underlying surface being isomorphic to Dn × Z2 , where Dn denotes the dihedral group of order 2n and Z2 is the group of order 2, for some even integer n. Also, the bound for the total number of ovals of the set of symmetries, in terms of g and n is proposed. By an extremal Riemann surface of even genus we shall understand a surface which attains the maximal number of 4 non-conjugate symmetries and the symmetries have the maximal possible total number of ovals—the bound is given in terms of g and n. If the surface of even genus has 4 conjugacy classes of symmetries, without any assumption on the number of ovals, then it shall be called s-extremal. In general, a Riemann surface shall be called s-extremal if it admits t
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