This book contains extended notes of most of the lectures given at the (instructional) Conference on Topics on Riemann Surfaces and Fuchsian Groups held in Madrid in 1998 to mark the 25th anniversary of the Universidad Nacional de Educación a Distancia. We wish to thank all contributors for their talks and texts, and in a special way to Alan F. Beardon for the excellent introduction. All papers have been refereed and we also thank to the referees for their work.

On combinatorial approach in studies of automorphisms of Riemann surfaces

Throughout the lecture a Riemann surface is meant to be a compact Riemann surface of genus g ≥ 2 and a symmetry of such surface, an antiholomorphic involution. The reader who has attended the lecture must be convinced of the importance of studies of symmetries of Riemann surfaces and the role that the groups of their automorphisms play there. Up to certain extent that lecture illustrates how to use results concerning this subject and in this one we shall mainly show how to get them. A symmetric Riemann surface X corresponds to a complex curve CX which can be defined over the reals and symmetries nonconjugated in the group Aut±(X) of all automorphisms of X correspond to nonequivalent real forms of CX. The aim of this lecture is a brief introduction to the combinatorial aspects of this theory together with samples of results and proofs. The most natural questions that arise here are the following:

• does a Riemann surface X admit a symmetry?

• how many nonconjugated symmetries may a given Riemann surface X admit?

• what can one say about the topology of these symmetries, e.g. about the number of their ovals or about their separability?

Riemann surfaces form a category with the holomorphic maps as morphisms and this category is closed under the quotients with respect to the action of groups of holomorphic automorphisms.

Introduction. Belyi's Theorem and the associated theory of dessins d'enfants has recently played an important rôle in Galois theory, combinatorics and Riemann surfaces. In this mainly expository article we describe some consequences for Riemann surface theory. It is organised as follows: in §1 we describe the ideas of critical points and critical values which leads in §2 to the definition of a Belyi function. In §3 we state Belyi's Theorem and define a Belyi surface. In §4 the close connection with triangle groups is described and this leads in §5 to an account of maps and hypermaps (or dessins d'enfants) on a surface. These are closely related to Belyi surfaces but in order to investigate this connection we introduce the idea of a smooth Belyi surface and a platonic surface in §6. The only new result in this article is Theorem 7.1 which describes the connection between regular maps and thir underlying Riemann sufaces.

Preliminaries

The definition of a Riemann surface is given in Beardon's lecture in this volume and the important ideas of uniformization are also discussed there. In the definition of a Riemann surface the transition functions are complex analytic and this allows us to describe most of the important ideas of complex analysis on a Riemann surface.

In this chapter, we will study eigenvalues and eigenfunctions for the Laplacian Δμ associated with (D, r) and μ. In particular, we will be interested in the asymptotic behavior of the eigenvalue counting function and present a Weyl-type result (Theorem 4.1.5) in 4.1.

It turns out that the nature of eigenvalues and eigenfunctions of Δμ is quite different from that of Laplacians on a bounded domain of ℝn. For example, we will find localized eigenfunctions in certain cases. More precisely, in 4.3, we will define the notion of pre-localized eigenfunctions, which are the eigenfunction of Δμ satisfying both Neumann and Dirichlet boundary conditions. It is known that such an eigenfunction does not exists for the ordinary Laplacian on a bounded domain of ℝn. Proposition 4.3.3 shows that if there exists a pre-localized eigenfunction, then, for any open set O ⊆ K, there exists a pre-localized eigenfunction whose support is contained in O.

One important consequence of the existence of pre-localized eigenfunctions is the discontinuity of the integrated density of states. See Theorem 4.3.4 and the remark after it.

We will give a sufficient condition for the existence of pre-localized eigenfunctions in 4.4. In particular, we will see that there exists a pre-localized eigenfunction for the Laplacian on an affine nested fractal associated with the harmonic structure appearing in Theorem 3.8.10. See Corollary 4.4.11.

In this chapter, we will discuss limits of discrete Laplacians (or equivalently Dirichlet forms) on a increasing sequence of finite sets. The results in this chapter will play a fundamental role in constructing a Laplacian (or equivalently a Dirichlet form) on certain self-similar sets in the next chapter, where we will approximate a self-similar set by an increasing sequence of finite sets and then construct a Laplacian on the self-similar set by taking a limit of the Laplacians on the finite sets.

More precisely, we will define a Dirichlet form and a Laplacian on a finite set in 2.1. The key idea is that every Dirichlet form on a finite set can be associated with an electrical network consisting of resistors. From such a point of view, we will introduce the important notion of effective resistance. In 2.2, we will study a limit of a “compatible” sequence of Dirichlet forms on increasing finite sets. Roughly speaking, the word “compatible” means that the Dirichlet forms appearing in the sequence induce the same effective resistance on the union of the increasing finite sets. In 2.3 and 2.4, we will present further properties of limits of compatible sequences of Dirichlet forms.

In this chapter, we will construct the analysis associated with Laplacians on connected post critically finite self-similar structures. In this chapter, L = (K, S, {Fi} i∊S) is a post critically finite (p. c. f. for short) self-similar structure and K is assumed to be connected. (Also in this chapter, we always set S = {1,2, …, N }.) Recall that a condition for K being connected was given in 1.6.

The key idea of constructing a Laplacian (or a Dirichlet form) on K is finding a “self-similar” compatible sequence of r-networks on {Vm } m≥0, where Vm = Vm(L) was defined in Lemma 1.3.11. Note that {Vm } m≥0 is a monotone increasing sequence of finite sets. We will formulate such a self-similar compatible sequence in 3.1. Once we get such a sequence, we can use the general theory in the last chapter and construct a resistance form (ℇ, F) and a resistance metric R on V∗, where V>∗ = Um≥0 Vm.

If the closure of V* with respect to the metric R were always identified with K, then we could apply Theorem 2.4.2 and see that (ℇ, F) is a regular local Dirichlet form on L2 (K, μ) for any self-similar measure μ on K. Consequently, we could immediately obtain a Laplacian associated with the Dirichlet form (ℇ, F) on L2(K, μ).