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The Engaging Symmetry of Riemann Surfaces: a Historical Perspective

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The Engaging Symmetry of Riemann Surfaces: a Historical Perspective Rose-Hulman Institute of Technology From the SelectedWorks of S. Allen Broughton April 27, 2021 The ne gaging symmetry of Riemann surfaces: a historical perspective Sean A Broughton Gareth A. Jones, University of Southampton David Singerman, University of Southampton Available at: https://works.bepress.com/allen_broughton/102/ Contemporary Mathematics The engaging symmetry of Riemann surfaces: a historical perspective S. Allen Broughton, Gareth A. Jones, and David Singerman Abstract. The historical development of Riemann surfaces, starting in the late 1800's, was driven in large part by the study of highly symmetrical sur- faces. Not only do these surfaces have an engaging beauty, but they have very strong interconnections with other structures such as maps on a surface. In this expository article we first develop the basics of Riemann surfaces and their automorphism groups, laying out the tools for the historical treatment of the highly symmetrical surfaces in the later sections. The main topics of the later sections will be Hurwitz surfaces and groups, maps and dessins d'enfant on surfaces, and platonic and quasi-platonic surfaces and groups. For the novice reader, the introductory material will also be helpful background for reading some of the other papers in this volume, particularly the companion article on future directions in the field. 1. Introduction As described in the preface, this volume focuses on the interests of a loose inter- national collaboration of over 100 researchers, especially in a series of conferences and special sessions over the past 3 decades. In this expository article we explore several areas of interest with a central theme of a compact Riemann surface with automorphisms. The investigation of automorphisms of Riemann surfaces began in the late 1800's, but, much to the pleasure of the authors, the field continues to develop to this day with many new techniques, new questions and new applications. After developing the basics from a fairly modern perspective, we give a historical per- spective on the main focus of this paper which is geometrical structures on highly symmetrical surfaces. It is these intriguing ideas that have sustained interest in this study and propel continuing research. It is very satisfying { perhaps contributing to over 100 years of continuing interest { that the topic draws from so many areas of mathematics: geometry/topology, algebra, combinatorics, analysis, number theory, and Galois theory. 1991 Mathematics Subject Classification. Primary: 14H37, 14H57, 30F10. Key words and phrases. Riemann surface, automorphisms, triangle group, dessin d'enfant. c 0000 (copyright holder) 1 2 BROUGHTON, JONES, AND SINGERMAN In light of this interplay, and due to the limited amount of space for this article, we are going to focus primarily on surfaces where the group has a significant impact on the geometric structures and other structures associated to the surface. The interplay among all the areas of mathematics is manifested in the following two fundamental and very deep theorems about Riemann surfaces. The first is that there is an equivalence between the categories of compact Riemann surfaces and of smooth projective algebraic curves and their functions fields. The second is the uniformisation theorem, that the only simply-connected Riemann surfaces are the Riemann sphere, the complex plane and the hyperbolic plane. These three surfaces come with a natural metric, the spherical, Euclidean or hyperbolic metric. The universal covering surface of any Riemann surface is one of these three. However the sphere can cover only itself, and the complex plane can cover only itself, a torus or the punctured plane, so all other Riemann surfaces have the upper half plane H (conformally equivalent to the open unit disc) as their universal covering surface. It follows that all these surfaces can be represented as H=Γ where Γ is a discrete group of hyperbolic isometries, that is, a Fuchsian group. We shall explore all these ideas in the preliminary sections. Acknowledgment. The authors wish to thank the referees for their hard work and excellent suggestions for the improvement of this article. 1.1. Overview of the sections. In Section2 we introduce the main actor, a Riemann surface with conformal automorphisms, and develop its character. After defining Riemann surfaces in several equivalent forms, we describe tools for con- struction and analysis of surfaces with automorphisms: including defining equa- tions, branched covers, the Riemann Existence Theorem, the Riemann-Hurwitz formula and uniformisation by Fuchsian groups. Using these tools, very significant progress has been made on the fascinating, widely studied topic of classification of surfaces with automorphisms, especially in low genus. The connection between the automorphism group and the geometry of a surface is strong when the group is large in comparison to the genus. Perhaps the most interesting class of highly symmet- rical surfaces are the Hurwitz surfaces, which are surfaces of maximal symmetry. We examine these surfaces and other similar families in Section3. Given any manifold with a group of automorphisms we may consider the fol- lowing problem. Is there a geometrical structure on the manifold that is preserved by the group, and does the geometrical structure generate the group structure in some way. Think of the symmetries of a soccer ball. For Riemann surfaces there is a very rich theory of dessins d'enfant, Bely˘ıfunctions, and maps on surfaces, which bring together three interesting concepts: 1) existence of a map or hypermap on the surface, 2) rigidity of the automorphism group (for regular surfaces), and 3) being defined over a number field. This topic, started by Grothedieck, combines many areas of mathematics: geometry and combinatorics of maps and hypermaps, number theory and Galois theory. We discuss these topics in Sections4,5 and6. For details about other closely related topics we do not discuss in this paper, we defer to other papers in this volume. For example, see [9] for a discussion of mod- uli spaces, [8] for topological aspects of actions and higher order differentials, and [79] for a discussion of super-elliptic curves. The companion article [10] on cur- rent open problems in the field, also gives some background on moduli space and Teichm¨ullerspace, genus spectrum, mapping class groups, and symmetries. For SYMMETRY OF SURFACES 3 the closely related topics of symmetries of Riemann surfaces and automorphisms of Klein surfaces see [11] and [12]. A word about notation. Certain topics in Riemann surfaces have established notation. While we have attempted to coordinate notation between the sections, we have given due deference to established notation. 2. Compact Riemann surfaces and their automorphisms 2.1. Compact Riemann surfaces. We shall employ several approaches to describing, constructing, and analysing Riemann surfaces and their automorphisms. In the following list, we just briefly describe each approach, and will add details later as needed. A good background reference for Riemann surfaces is [28]. (1) Complex one dimensional manifold A compact Riemann surface S is a compact one-dimensional complex manifold. Specifically S has a covering by open sets fUi : i 2 Ig ; and an atlas of bijective coordinate chart homeomorphisms φi : Ui ! C such that for every pair i; j the map −1 φj ◦ φi : φi(Ui \ Uj) $ φj(Ui \ Uj) is a biholomorphic homeomorphism of the opens sets φi(Ui \ Uj) and φj(Ui \ Uj) in C: This definition is the correct viewpoint to do analysis on surfaces, define holomorphic and meromorphic functions on the surface, and define holomorphic maps from one surface to the other. (2) Smooth projective curve, defining equations A compact Riemann surface S is a smooth irreducible projective curve. i.e., S is the set of n common zeros in P (C) of a set of homogeneous polynomials, f1; : : : ; fs of n + 1 homogenous variables X0;:::;Xn: A typical way for this to happen is to have a single equation for an affine plane curve 2 (2.1) Sf = (x; y) 2 C : f(x; y) = 0 : To get a Riemann surface S we need to take the projective completion of 2 Sf in P (C) and then repair the singularities through normalization. The resulting curve may not lie in P2(C): The function f is called a defining equation and is often difficult to find, though of great interest. We could also look for f1; : : : ; fs: When we want to emphasize that S is defined by an algebraic equation we will call it a smooth complex curve. (3) Branched cover A branched cover π : S ! T is simply a holomor- phic map of surfaces in the sense of complex manifolds. Branched cov- ers are connected to meromorphic functions by noting that a meromor- phic function on S amounts to a branched cover π : S ! Cb: A typical branched cover may be constructed from a defining equation Sf = 0. De- fine πf : Sf ! C by (x; y) ! x and modify to a proper branched cover π : S ! Cb. In this paper we will only consider branched covers of the Rie- mann sphere Cb. We will give a informal construction of branched covers dating from the last century and link it to two important theorems: the Riemann Existence Theorem and the Riemann-Hurwitz formula. (4) Quotient of universal cover, uniformisation As previously stated, we can describe S as a quotient of its universal cover. The universal cover U will be one of Cb = P 1(C) the Riemann sphere, C the complex plane or, H the hyperbolic plane. There will be a properly discontinuous group of automorphisms Π ' π1(S) such that U ! U=Π ' S is a holomorphic 4 BROUGHTON, JONES, AND SINGERMAN covering space of manifolds. We call the three different possibilities spher- ical, Euclidean and hyperbolic to reflect the underlying geometry. In the spherical case Π is trivial and Π ≈ Z2 in the Euclidean case.
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