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Graduate Center, City University of New York

This work is made publicly available by the City University of New York (CUNY).

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Chapter 1 Introduction

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A half-translation structure on a finite Riemann surface is an atlas of conformal charts (for the surface minus a finite set) whose transition functions are of the form z 7→ ±z + c (with some extra condition on the finite set) and in which the surface has finite area. This data can be specified by an integrable holomorphic quadratic differential. The charts of the half-translation structure should also map the ideal boundary to curves which are piecewise

1Since X is locally compact Hausdorff, a path [0, 1] → Map(X, Y ) is the same as a homotopy X×[0, 1] → Y [Mun00, p.287].

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into Y . Montel’s theorem in complex analysis implies that:

2If not, then for any f ∈ CEmb(X, Y, h) the complement Y \ f (X) is finite so that f is a slit mapping, as long as Y supports a half-translation structure. The only finite Riemann surface with non-abelian funda- mental group which does not support any half-translation structure is the triply punctured sphere. But if each of X and Y is a punctured sphere and h is generic, then CEmb(X, Y, h) has at most one element, so there is nothing to show.

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(Y ), and X is the complement of a collection

2This case is not relevant here since we assume Y to be hyperbolic, but nevertheless.

3The source of the mistake is [Iof75, Lemma 3.2]. Similarly, [Iof78], [EM78], and [GG01] contain minor errors as they build up on the false claim.

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