On compact Riemann surfaces with dihedral groups of automorphisms (original) (raw)
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Analytic computation of some automorphism groups of Riemann surfaces
Kodai Mathematical Journal, 2007
Equations for the locus of Riemann Surfaces of genus three with a nonabelian automorphism group generated by involutions are determined from vanishings of Riemann's theta function. Torelli's Theorem implies that all of the properties of a non-hyperelliptic compact Riemann Surface (complex algebraic curve) X are determined by its period matrix W. This paper shows how to compute the group Aut X of conformal automorphisms of a surface X of genus three using W, in the case when the group is nonabelian and generated by its involutions. The connection between W and X is Riemann's theta function yðz; WÞ. Accola ([1], [2], [3]), building on classical results about hyperelliptic surfaces, found relationships between the theta divisor Y ¼ fz A JacðX Þ : yðz; WÞ ¼ 0g and Aut X. In the case of genus three, certain vanishings of y at quarter-periods of JacðX Þ imply that X has an automorphism s of degree two (or involution) such that X =hsi has genus one (making s an elliptic-hyperelliptic involution). This work derives equations in the moduli space of surfaces of genus three for many of the loci consisting of surfaces with a given automorphism group. It is a two-step process. First, topological arguments determine the order of the dihedral group generated by two non-commuting involutions. Then, combinatorial arguments about larger groups generated by involutions determine the theta vanishings corresponding to each. Much of the work here is based on the author's 1981 PhD dissertation [7] at Brown University. It appears now because of renewed interest in these questions, some of which is inspired by questions in coding theory: See [3], [5]. The research was directed by R. D. M. Accola, and Joe Harris was also a valuable resource. The author extends his (belated) thanks to them. 1. Preliminaries and notation In all that follows, X is a compact Riemann Surface (or complex algebraic curve) of genus three with automorphism group Aut X , period matrix W, jacobian 394
Symmetries of Riemann surfaces with large automorphism group
Mathematische Annalen, 1974
A Riemann surface is symmetric if it admits an anti-conformal involution. The basic question which we discuss in this paper is whether compact Riemann surfaces of genus g > t which admit large groups of automorphisms are symmetric. As is weU-known, the automorphism group of a compact Riemann surface of genus g > 1 is finite and bounded above by 84(g-1). Macbeath ([t21 13]) has found infinitely many g for which this bound is attained. We show that all the surfaces found by Macbeath's methods are indeed symmetric. However, we do exhibit an example of a non-symmetric Riemann surface of genus g = ! 7 which does admit 84(g-1) automorphisms. We also study Riemann surfaces admitting automorphisms of large order. The order of an automorphism of a Riemann surface of genus g is bounded above by 4g + 2 and this bound is attained for every g [8]. We show that all Riemann surfaces admitting automorphisms of order greater that 2g + 2 are symmetric. There is a close link between our work and the theory of irreflexible regular maps on surfaces. (See § 8 for definitions.) There is a connection between the groups of regular maps and large groups of automorphisms of compact Riemann surfaces. Indeed, every group of automorphisms ofa Riemann surface of genus g of order greater than 24(g-1) is also the group of some regular map and conversely, every group of a regular map can be thought of as the group of automorphisms of a Riemann surface. The irreflexible regular maps turn out to be rather exceptional. (In fact, it was suggested in early editions of [3] that they did not exist for surfaces of genus O > 1). We show in the above correspondence that large groups of automorphisms of non-symmetric surfaces will give rise to irreflexible regular maps, but that the converse of this fact is not always true. Thus, for example, groups of automorphisms of order greater than 24(g-1) of a compact non-symmetric Riemann surface of genus g are more exceptional than irreflexible regular maps. There is another interpretation of symmetric Riemann surfaces which is of interest. Every compact Riemann surface can be obtained as the Riemann surface of an algebraic curve f(z, w) = 0. A Riemann surface
One-dimensional families of Riemann surfaces of genus g with 4\text {g}+4$$ 4 g + 4 automorphims
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017
We prove that the maximal number ag + b of automorphisms of equisymmetric and complex-uniparametric families of Riemann surfaces appearing in all genera is 4g + 4. For each integer g ≥ 2 we find an equisymmetric complex-uniparametric family A g of Riemann surfaces of genus g having automorphism group of order 4g + 4. For g ≡ −1mod4 we present another uniparametric family K g with automorphism group of order 4g + 4. The family A g contains the Accola-Maclachlan surface and the family K g contains the Kulkarni surface.
Poincaré's theorem for the modular group of real Riemann surfaces
Differential Geometry and its Applications, 2009
Let M od g denote the modular group of (closed and orientable) surfaces S of genus g. Each element [h] ∈ M od g induces a symplectic automorphism H([h]) of H 1 (S, Z). Poincaré showed that H : M od g → Sp(2g, Z) is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution σ. Let (S, σ) be a real Riemann surface, Homeo σ g be the group of orientation preserving homeomorphisms of S such that h • σ = σ • h and Homeo σ g,0 be the subgroup of Homeo σ g consisting of those isotopic to the identity. The group M od σ g = Homeo σ g /Homeo σ g,0 plays the rôle of the modular group in the theory of real Riemann surfaces. In this work we describe the image by H of M od σ g. Such image depends on the topological type of the involution σ.
2004
We construct a special type of fundamental regions for any Fuchsian group FFF generated, by an even number of half-turns, and for certain non-Euclidean crystallographic groups (NEC groups in short). By comparing these regions we give geometrical conditions for F to be the canonical Fuchsian subgroup of one of those NEC groups. Precisely speaking, we deal with NEC groups of algebraic genus 0 having all periods in the signature equal to 2. By means of these conditions we give a characterization of hyperelliptic and symmetric Riemann surfaces.
Triangulations and moduli spaces of Riemann surfaces with group actions
Manuscripta Mathematica, 1995
We study that subset of the moduli space Ma of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite group F acts. We show first that this subset is compact. It turns out that, for general finite groups F, the above subset is not connected. We show, however, that for Z2 actions this subset is connected. Finally, we show that even in the moduli space of smooth genus g Riemann surfaces, the subset of those Riemann surfaces on which Z2 acts is connected, ha view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certain regular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in apl)roximating eigenfunctions of tim Laplace operator numerically.
Riemann surface with cyclic automorphisms group
Proyecciones (Antofagasta), 1997
In t his paper. we present tllC' uniformization of y 2 = .rP-l, with p > 5 aurl prime. i. e .. the only hyperelliptic Riemann surface of gt'nus (/-7. \\"hich admit Z j2pZ as automorphism group. This 1111ifonnization is fouud by using a fuc:hsian group which rcflects the actiou of Z/2pZ aud is coustructed starting of a triangle group of !YJW (0:¡>.p.p). I\loreover. we describe completely the action of the automorphism group in hmnology. so that we can describe the invariant subvariety for Z /2pZ in A 9 (principally polarized abelian varieties of dimension y). which is detPrmiued bv the real Abe! aplication from M 9 in A 9 .
Automorphisms of compact non-orientable Riemann surfaces
Glasgow Mathematical Journal, 1971
Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infin...
On Riemann surfaces of genus g with 4g–4 automorphisms
Israel Journal of Mathematics
In this article we study compact Riemann surfaces with a nonlarge group of automorphisms of maximal order; namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g − 4. Under the assumption that g − 1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.