Fixed Points on Asymmetric Riemann Surfaces
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Fixed Points on Asymmetric Riemann Surfaces Ewa Kozlowska-Walania and Ewa Tyszkowska Abstract. We study actions of cyclic groups on asymmetric Riemann surfaces for which all conformal automorphisms of prime orders have the same number of fixed points. We find the sharp upper bound on the number of points fixed by the square of an anticonformal automorphism of an asymmetric surface. Moreover, we determine the minimal genus of an asymmetric Riemann surface on which a given finite group G acts without fixed points. Mathematics Subject Classification. Primary 30F99; Secondary 14H37, 20F. Keywords. Riemann surface, symmetry of Riemann surface, asymmetric Riemann surface, pseudo-symmetric Riemann surface, Fuchsian groups, NEC groups, fixed points on Riemann surfaces, fixed point free actions on Riemann surfaces.
1. Introduction In this paper, we study the so-called asymmetric (or pseudo-real, see [1]) Riemann surfaces. This term means that the surface admits orientation reversing automorphisms, but none of them is an involution. Such automorphisms are called asymmetries and they have orders divisible by 4. An asymmetry of order 4 is called a pseudo-symmetry and the underlying Riemann surface is called pseudo-symmetric. Let us also mention that an orientation reversing involution is called a symmetry, and the surface on which it acts is called symmetric. The category of complex algebraic curves is equivalent to the category of compact Riemann surfaces. It is known that the moduli space Mg of complex algebraic curves of genus g is a quasi-projective variety which can be defined in Pn (C) by polynomials with rational coefficients. There is an antiholomorphic involution ι : Mg → Mg which maps the class of a complex Ewa Kozlowska-Walania was supported by Polish National Sciences Center by the grant NCN 2015/17/B/ST1/03235.
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curve to its conjugate. The fixed points of such a mapping are called complex algebraic curves with real moduli. The defining equations of symmetric Riemann surfaces are over reals and therefore they are fixed by ι. The other fixed points, which are not symmetric, correspond to the asymmetric surfaces. There are many publications concerning asymmetric and pseudosymmetric Riemann surfaces, let us only mention the most important, from our point of view, which were the motivation for this work. Many general results concerning asymmetric surfaces can be found in [1] and some asymmetric surfaces with cyclic automorphism groups were also considered in [4]. The hyperelliptic asymmetric surfaces were studied in [12] and in [2], where the defining equations for such surfaces are given and the special case of hyperelliptic pseudo-symmetric surfaces is also treated. The present paper can be seen as a continuation of the authors’ papers [8,9,13], where asymmetric p-hyperelliptic and (q, n)-gonal surfaces, with cyclic automorphism group, were studied. We shall also recall the two papers concerning fixed points on Riemann surfaces, which are rel
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