Finite branched covers of the unit disk

Abstract

Following recent work of several authors, this thesis continues the study of hyperbolic Belyi maps and investigates the possibility for formulating a hyperbolic analogue of Belyi's theorem. \par A hyperbolic Belyi map is a finite branched cover $f:X\rightarrow \disk$ with at most two critical values, where $X$ is a Riemann surface. For $X\cong \disk$, it can be represented by what is called Shabat-Blaschke products. They are characterized by the combinatorial data encoding the bicolored graph formed by the preimage of the hyperbolic geodesic segment joining the two critical values. When $X\cong \disk$, the underlying graph is a tree. A particular kind of examples is the family of Chebyshev-Blaschke products, whose graphs are the $n$-chains. They have been studied extensively by some recent authors, and form a major inspiration of this thesis. \par With Chebyshev-Blaschke products, the prototypical example in mind, we then consider a Riemann surface $X$ with a finite branched cover $f: X\rightarrow \disk$. We show that there exists a compact bordered Riemann surface $\overline{X}$ containing $X$ as its interior, on which all finite branched covers from $X$ to $\disk$ can be extended continuously to the boundary. With $\overline{X}$, we double it to obtain a compact Riemann surface $\overline{X}_D$ with an antiholomorphic involution $\Phi$, such that any finite branched cover $f:X\rightarrow \disk$ is extended to meromorphic function $\tilde{f}$ that satisfy the functional property $\tilde{f}\circ \Phi=\frac{1}{\overline{\tilde{f}}}$. We give a condition for a compact Riemann surface with an antiholomorphic involution to be coming from some compact bordered Riemann surface (Corollary 2.12). We classify all bordered Riemann surfaces whose double are of genus one or two (Proposition 2.17 and Theorem 2.18). In each case, we give a complete description of when a meromorphic function on the double of a compact bordered Riemann surface comes from extending a finite branched cover from the original compact bordered Riemann surface to $\overline{D}$. Given a compact bordered Riemann surface $X$ with a finite branched cover $f:X\rightarrow \overline{\disk}$, we also consider the problem of finding a compact Riemann surface $Y$ and an embedding $\phi:X\rightarrow Y$ such that all finite branched covers from $X$ to $\overline{\disk}$ can be extended (as functions of $\phi(X)$) to meromorphic functions of $Y$. For the case of annuli, whose double are complex tori, we show there does not exist such embedding into $\mathbb{P}^1$ (Proposition 2.27). \par Moreover, we give examples of family of finite branched cover to $\disk$ with a given monodromy, represented by their double. We revisit the Chebyshev-Blaschke product and find another parametrization of the family of degree three Chebyshev-Blaschke products, which converges to degree three Chebyshev polynomials. We give further examples of families of Shabat-Blaschke products with tree monodromies. Finally, an additional simple example of a family of hyperbolic Belyi maps, whose double are meromorphic functions on complex tori, is given. Its convergence to ordinary Belyi map is demonstrated. We also prove a condition for a Shabat-Blaschke product $f$ to be equivalent to an odd function, in the sense that there exists $\omega\in \text{Aut}(\disk)$ such that $f\circ \omega$ is an odd function (Proposition 3.1).