On n-th roots of meromorphic maps

Juan C García,Rubén A Hidalgo

doi:10.15446/recolma.v54n1.89789

Abstract

Let S be a connected Riemann surface and let φ: S → Ĉ bebranched covering map of nite type. If n ≥ 2,then we describe a simple geometrical necessary and sucient condition for the existence of some n-th root, that is, a meromorphic map ψ: S → Ĉ such that φ = ψn.

Highlights

In this paper, S will denote a connected Riemann surface and C = C ∪ {∞} will be the Riemann sphere
We say that φ is of finite type if the sets Mq are finite
Let K be a finitely generated Fuchsian group, acting on the hyperbolic plane H2, such that H2/K is an orbifold of genus zero and let k1, . . . , kr be the orders of its cone points

Summary

Introduction

The existence of an n-th root of φ necessarily implies that: (a) ∞, 0 ∈ Bφ and (b) the branch orders of both 0 and ∞ are multiples of n These two conditions are not sufficient for φ to have an n-root. Let K be a finitely generated Fuchsian group, acting on the hyperbolic plane H2, such that H2/K is an orbifold of genus zero (so its underlying Riemann surface structure is C) and let k1, . The uniformization theorem asserts that, for a connected Riemann surface S of hyperbolic type, each branched covering of finite type φ : S → C is obtained in such a way for suitable choices of Γ and K. Main results Before stating our main result we need some definitions

Admissible arcs and n-Z-orientability

Case of compact Riemann surfaces

A remark: θ-Zapponi-orientability of Kleinian groups