Discrete Groups and Riemann Surfaces Anthony Weaver July 6, 2013 Abstract These notes summarize four expository lectures delivered at the Ad- vanced School of the ICTS Program Groups, Geometry and Dynamics, De- cember, 2012, Almora, India. The target audience was a group of students at or near the end of a traditional undergraduate math major. My purpose was to expose the types of discrete groups that arise in connection with Riemann surfaces. I have not hesitated to shorten or omit proofs, espe- cially in the later sections, where I thought completeness would interrupt the narrative flow. References and a guide to the literature are provided for the reader who demands all the details. Contents 1 Prerequisites 2 2 Riemann surfaces 2 2.1 The Riemann sphere . 3 2.2 The graph of an analytic function . 3 2.3 Smooth affine plane curves . 3 2.4 Smooth projective plane curves . 4 3 Holomorphic maps 5 3.1 Automorphisms . 6 3.2 Meromorphic functions . 6 3.3 The local normal form . 7 3.4 The Riemann-Hurwitz relation . 8 3.5 Fermat curves . 9 3.6 Cyclic covers of the line . 10 3.7 Resolving singularities . 10 4 Galois groups 12 4.1 The monodromy group . 12 4.2 Two permutation groups . 14 4.3 Galois coverings . 15 1 4.4 A presentation for the Galois group . 16 4.5 The dihedral group as a Galois group . 17 4.6 Galois coverings of the line . 18 4.7 The Galois extension problem . 18 5 Uniformization 19 5.1 The Dirichlet region . 21 5.2 Surface groups . 25 5.3 Triangle groups . 25 5.4 Automorphisms via uniformization . 26 5.5 Surface-kernel epimorphisms . 27 5.6 Topological conjugacy . 28 5.7 Teichmüller spaces . 28 6 Greenberg-Singerman extensions 30 6.1 Generalized Lefschetz curves . 33 6.2 Accola-Maclachlan and Kulkarni curves . 35 7 Further reading 35 1 Prerequisites To progress beyond the definition of a Riemann surface, one needs to know a little bit about a lot of things. Accordingly, here are the prerequisites necessary to begin these notes. (i) Complex analysis: analytic functions, conformal map- pings, Taylor series as in [4]. (ii) Topology: open sets, homeomorphisms, open mappings, the fundamental group, covering spaces as in [28]. (iii) Groups and group actions: permutation groups, normal subgroups, factor groups, isotropy subgroups, group presentations as in [38]. (iv) Hyperbolic geometry: the up- per half plane and disk models, the Gauss-Bonnet theorem as in [6]. We start from this broad baseline. I give a brief guide to further reading in the final section, for those readers whose appetite has been whetted by these brief notes. 2 Riemann surfaces A Riemann surface is an abstract object that, locally, looks like an open subset of the complex plane C. This means one can do complex analysis in a neigh- borhood of any point. Globally, a Riemann surface may be very different from C, however. For example, it could be compact, and it need not be simply con- nected. Here is the technical definition. Definition 1. A Riemann surface X is a second-countable, connected, Haus- dorff space with an atlas of charts, φ : U V , where U , V are open ↵ ↵ ! ↵ ↵ ↵ 2 subsets of X, C, respectively, and φ↵ is a homeomorphism. For every pair of charts φ↵, φβ with overlapping domains, the transition map, 1 φ φ− : φ (U U ) φ (U U ) β ◦ ↵ ↵ ↵ \ β ! β ↵ \ β is bianalytic, that is, analytic with analytic inverse. Some basic examples follow. 2.1 The Riemann sphere A two-chart atlas on S2 = (x, y, w) R3 x2 + y2 + w2 =1 is given by { 2 | } stereographic projection from the north and south poles: 2 x y φ1 : S (0, 0, 1) C, (x, y, w) + i \ ! 7! 1 w 1 w − − 2 x y φ2 : S (0, 0, 1) C, (x, y, w) i . \ − ! 7! 1+w − 1+w The inverses of these maps are 2 1 2Re(z) 2Im(z) z 1 φ− (z)= , , | | − 1 z 2 +1 z 2 +1 z 2 +1 ✓ ◆ | | | | | | 2 1 2Re(z) 2Im(z) 1 z φ− (z)= , − , −| | . 2 z 2 +1 z 2 +1 z 2 +1 ✓| | | | | | ◆ 1 The transition map φ φ− is simply z 1/z. 2 ◦ 1 7! 2.2 The graph of an analytic function For an analytic function w = g(z) whose domain contains the open set U C, 2 ✓ the graph (z,g(z)) z U C , with the single chart ⇡z :(z,g(z)) z, is a { | 2 }✓ 7! Riemann surface. 2.3 Smooth affine plane curves Definition 2. An affine plane curve X is the zero locus of a polynomial f(z,w) 2 C[z,w]. It is non-singular or smooth if, for all p =(a, b) X, the partial deriva- 2 tives fz(p) and fw(p) are not simultaneously zero. By the implicit function theorem, in a neighborhood of every point p on a smooth affine plane curve, at least one of the coordinates z,w is an analytic function of the other, depending on which partial derivative is =0. If f (p) = 6 w 6 0, there is an open set U containing p such that, for all q =(z,w) U, w = g(z), 2 an analytic function of z. Thus ⇡z : U C is a local chart. If, also, fz(p) =0, ! 6 there is an open set V containing p such that, for all q =(z,w) V , z = h(w), 2 3 an analytic function of w. Then ⇡w : V C is also a local chart. The transition ! functions, 1 ⇡ ⇡− : z g(z) w ◦ z 7! 1 ⇡ ⇡− : w h(w), z ◦ w 7! defined on ⇡ (U V ) and ⇡ (U V ), respectively, are, by construction, analytic. z \ w \ Thus a smooth affine plane curve, if connected, is a Riemann surface. Remark 1. Connectivity can be guaranteed by assuming the polynomial f(z,w) is irreducible, that is, not factorable into terms of positive degree. This is a stan- dard result in algebraic geometry which is beyond the scope of this paper. See [36]. 2.4 Smooth projective plane curves The one-dimensional subspaces of the vector space C3 are the ‘points’ of the complex projective plane P2. The span of (x, y, z) C3, (x, y, z) =(0, 0, 0), is 2 6 denoted [x : y : z]. For λ C, λ =0, 2 6 [x : y : z]=[λx : λy : λz]. x, y and z are homogeneous coordinates on P2: being defined only up to a com- mon scalar multiple, no coordinate takes on any "special" or fixed value. P2 is a complex manifold of dimension 2, covered by three sets, defined by x =0, 6 y =0, z =0, respectively. In homogenous coordinates, we may assume that 6 6 x 2 + y 2 + z 2 =1; in particular, that x , y , z 1. Thus P2 is compact. | | | | | | | | | | | | 3 Definition 3. A polynomial F (x, y, z) C is homogeneous if, for every λ C⇤, 2 2 F (λx, λy,λz)=λdF (x, y, z), where d is the degree of the polynomial. On P2, the value of a homogeneous polynomial F (x, y, z) is not well-defined, but the zero locus is. Definition 4. A projective plane curve X is the zero locus in P2 of a homogeneous polynomial F (x, y, z) C[x, y, z]. It is non-singular (smooth) if, there is no point 2 p =[x : y : z] X, at which all three partial derivatives @ F (p), @ F (p), and 2 x y @zF (p) vanish simultaneously. An affine plane curve f(x, y)=0can be “projectivized" (and thereby, com- pactified) by the following procedure: multiply each term of the defining poly- nomial f by a suitable power of a new variable z so that all terms have the same (minimal) degree. Then the affine portion of the projectivized curve corresponds to z =1, and the points at infinity correspond to z =0. Theorem 1. A nonsingular projective plane curve is a compact Riemann surface. 2 Proof. Let Ui = [x0 : x1 : x2] P xi =0 , i =0, 1, 2. (Up to a nonzero scalar { ✓ | 6 } factor, x =0is equivalent to x =1.) Let X be a smooth projective plane curve i 6 i 4 defined as the zero locus of the homogenous polynomial F (x0,x1,x2), and let X = X U . Each X is an affine plane curve, e.g., i \ i i 2 X0 = (a, b) C F (1,a,b)=0 . { 2 | } For a homogeneous polynomial F of degree d, 1 k F (x ,x ,...x )= x @ F. 0 1 k d i i i=0 X This is known as Euler’s formula, and it implies (exercise) that X is nonsin- gular if and only if each Xi is a smooth affine plane curve. Coordinate charts on Xi are ratios of homogeneous coordinates on X, and as such they are well- defined. For example, charts on X0 are x1/x0 or x2/x0, and charts on X2 are x0/x2 or x1/x2. Transition functions are readily seen to be holomorphic, e.g., near p X X , where x ,x =0, let z = φ = x /x and w = φ = x /x . 2 0 \ 1 0 1 6 1 1 0 2 2 1 Then 1 h(z) φ φ− : z [1 : z : h(z)] = w, 2 ◦ 1 7! 7! z where h(z) is a holomorphic function, and z =0, since p X . Connectivity is 6 2 1 required to make Xi (and hence X) a Riemann surface.